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briBaØabR&tKNitviTüa nig BaNiC¢kmµ

sinx f(x) x 1

ebI ebI

x 0 x 0

rkßasiTiæ 2008

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elak lwm qun elak Esn Bisidæ elak Titü em¨g elakRsI Tuy rINa elak RBwm suxnit elak pl b`unqay

GñkrcnaRkb nig bec©keTskMBüÚT&r

kBaØa lI KuNÑaka

GñkRtYtBinitüGkçaraviruTæ

elak lwm miK:sir © rkßasiTæi lwm pl:ún 2008

GarmÖkfa

esovePA dMeNa¼RsaylMhat´KNitviTüaEdlGñksikßakMBugkan´ enAkñúgéden¼xMJúáTánxitxMRsavRCav nigniBnæeLIgkñúgeKalbMNg TukCaÉksarRsavRCavsRmab´GñksikßaEdlmanbMNgcg´ec¼ cg´dwgGMBIemeronen¼[kan´Etc,as´ . enAkñúgesovePA en¼ánRbmUlpþMúnUvRbFanlMhat´ya¨geRcInnigman lkçN¼xusEbø[email protected]ña .RbFanlMhat´[email protected]úáTánxitxMeRCIserIs ya¨gsRmitsRmaMgbMputRBmTaMgeFIVdMeNa¼Rsayya¨gek,a¼k,ayEdl Gac[Gñksikßagayyl´nigqab´cgcaMGMBIviFIsaRsþeFIVdMeNa¼Rsay lMhat´[email protected] . b¨uEnþeTa¼Caya¨gNak¾eday kgV¼xatbec©keTs Kruekaslü nig kMhusGkçraviruTæRákdCaekItmaneLIgedayGectna CaBMuxaneLIy . GaRs&yehtuen¼xMJúáTCaGñkniBnæ rg´caMTTYlnUvmti ri¼Kn´EbbsSabnaBIsMNak´GñksikßakñúgRKb´mCÄdæanedaykþIesamnsßrIk rayCanic©edIm,IEklMGesovePAen¼[kan´EtmansuRkitüPaBEfmeTot . xMJúáTCaGñkniBnæsgÇwmfaesovePAsikßaGnuKmn_mYyk,alen¼ nwgcUlrYmnaMelakGñkeq<a¼eTArkC&yCMn¼kñúgkarsikßa nig kar RbLgRbECgnanaCaBMuxaneLIy . sUm[GñksikßaTaMgGs´mansuxPaBlðman®áCJaQøasév nigman sMNaglðkñúgqakCIvit nig karsikßa !

át´dMbgéf¶TI 22 Ex kumÖ¼ qñaM 2008 GñkniBnæ nig RsavRCav lwm pl:ún

emeronTI1

cMnYnkMupøic

1-niymn&y cMnYnEdlmanTRmg´ z a i.b Edl a nig b CacMnYnBit ehAfacMnYnkMupøic . TRmg´ z a i.b ehAfaTRmg´BICKNiténcMnYnkMupøic. a ehAfaEpñkBitEdleKkMnt´sresr Re(z) a . b ehAfaEpñkBitEdleKkMnt´sresr Im(z) b . i ehAfaÉktanimµitEdl i 1 rW i 1 eKtagsMNMucMnYnkMupøiceday ¢ . 2-RbmaNviFIelIcMnYnkMupøic ½ snµtfaeKmancMnYnkMupøic z a i.b nig z' a'i.b' Edl a ; b ; a' ; b' CacMnYnBit . -plbUk z z' (a a') i.(b b') -pldk z z' (a a') i(b b') -plKuN z z' (aa'bb') i(ab'a' b)

2

1

-cMras´ 1 aa i.bb z aa -plEck zz' a 'bb' i. ab'ab' b a b 3-ÉklkçN¼PaBKYrkt´sMKal´ ebI z nig z' CaBIrcMnYnkMupøicena¼eKman ½

2 2 2 2 2 2

(z z' ) 2 z 2 2z z' z'2 z 2 z' 2 (z z' )(z z' )

;

(z z' ) 2 z 2 2z z' z'2

1 z n 1 1 z z ... z ; z 1 , n IN * 1 z

2 n n k (z z ' ) C n z k z' n k k 0 n

; n IN *

4-cMnYnkMupøicqøas´ k-ebIeKmancMnYnkMupøic z a i.b Edl a nig CacMnYnBit . cMnYnkMupøicqøas´én z tageday z a i.b . x-lkçN¼ ebI z nig z' CaBIrcMnYnkMupøicena¼eKman ½

z z ' z z' z z ' z z' ; z z' z z ' z z ; z' z'

b

2

K-sMKal´ ebI z a i.b ena¼ z a i.b eKán Re(z) z z nig Im(z) z 2i z 2 5-sV&yKuNén i cMeBa¼RKb´ k IN * eKman ½

i 4 k 1 ; i 4 k 1 i ; i 4 k 2 1 ; i 4 k 3 i

.

6-dMeNa¼RsaysmIkardWeRkTIBIr eK[smIkardWeRkTIBIr az bz c 0 ; a 0 , a; b; c IR -ebI b 4ac 0 smIkarman¦sBIrepßgKñaCacMnYnBitkMnt´eday b b z ; z . 2a 2a -ebI b 4ac 0 smIkarman¦sDubmYyCacMnYnBitkMnt´eday b z z . 2a -ebI b 4ac 0 smIkarman¦BIrCacMnYnkMupøicqøas´KñakMnt´eday b i || bi || z ; z . 2a 2a

2 2 1 2 2 1 2 2 1 2

3

7-kartagcMnYnkMupøictamEbbFrNImaRt k- cMnuc M énbøg´RbkbedaytRmuyGrtUnrm¨al´ (O, i , j) Edlman kUGredaen (a ; b ) ehAfacMnucrUbPaBén cMnYnkMupøic z a i.b . cMnYnkMupøic z a i.b ehAfaGahVikéncMnuc M(a ; b ) .

y

M

1

x 0 1

x-viucTr& W énbøg´RbkbedaytRmuyGrtUnrm¨al´ (O, i , j) Edlman kUGredaen (a ; b ) ehAfavicTr&rUbPaBén cMnYnkMupøic z a i.b . cMnYnkMupøic z a i.b ehAfaGahVikénviucTr& W(a ; b ) .

4

y

M

1

x 0 1

8-m¨UDuléncMnYnkMupøicmYy

y

b

M

1

0

1

a

x

k-ebIeKmancMnYnkMupøic z a i.b Edl a nig b CacMnYnBitm¨UDulécMnYnkMupøicen¼KWCacm¶ay OM kMnt´eday ½ | z | OM a b .

2 2

5

x-lkçN¼

| z | zz ; | z z '| | z | | z'| ; | z |z| | z' | z '|

.

K-vismPaBRtIekaN cMeBa¼RKb´cMnYnkMupøic

| z z' | | z | | z ' |

z

nig

z'

eKman ½

. X-cm¶ayrvagBIrcMnuckñúgbøg´ ebI A nig B CaBIrcMnucmanGahVikerogKña z nig z énbøg´RbkbedaytRmuyGrtUnrm¨al´ (O, i , j) eKán d(AB) | z z | . 9-TRmg´RtIekaNmaRténcMnYnkMupøicmYy

A

B

B

A

y

b

M

1

0

1

a

x

6

eKmancMnYnkMupøic z a i.b Edl a nig b CaBIrcMnYnBit TRmg´RtIekaNmaRtén z kMnt´eday

z r (cos i. sin )

Edl r a b nig cos ar ; sin br . 10-RbmaNviFIelIcMnYnkMupøicTRmg´RtIekaNmaRt snµtfaeKmancMnYnkMupøicRtIekaNmaRtBIr ½ z r (cos i.sin ) nig z ' r ' (cos 'i.sin ' ) -plKuN z z' r.r' cos( ' ) i.sin( ') -plEck zz' rr' cos( ') i.sin( ' ) 11-sV&yKuNTI n snµtfaeKmancMnYnkMupøicRtIekaNmaRt z r (cos i.sin ) sV&yKuNTI n éncMnYnkMupøicen¼kMnt´eday ½ z r (cos i.sin ) r cos( n) i.sin( n) RKb´ n Z 12-rUbmnþdWm&r cMeBa¼RKb´ IR ; n Z eKman ½ cos i.sin cos(n) i.sin( n) ( ehAfarUbmnþdWmr& )

2 2

n n n n

7

13-¦sTI n éncMnYnkMupøicmYy ebIeKman z r (cos i.sin ) CacMnYnkMupøicminsUnü nig n IN .¦sTI n éncMnYnkMupøicen¼kMNt´eday ½

2k 2k Wk n r cos( ) i.sin( ) ; k 0 , 1 , 2 ,...,(n 1). n n

14-TRmg´Giucs,¨ÚNg´EsüléncMnYnkMupøicmYy ebI z CacMnYnkMupøicmanm¨UDúl r nigGaKuym¨g´ ena¼TRmg´Giucs,¨ÚNg´Esülén z kMnt´eday z r.e 15-rUbmnþGWEl cMeBa¼RKb´cMnYnBit eKmanTMnak´TMng ½ e e cos nig sin e ie . 2 2 16-TMnak´TMngrvagTRmg´BICKNit-RtIekaNmaRt nig Giucs,¨ÚNg´Esül ½

i . i . i . i .

i.

z a i.b r (cos i. sin ) r.ei.

Edl

r a 2 b 2 ; cos

a b ; sin r r

.

8

lMhat;TI1 eK[cMnYnkMupøic a 2 3i ; b 3 i nig c 1 4i k-cUrsresr a b c nig a b c CaTRmg´BICKNit . x-cUrepÞógpÞat´faeKGackMnt´cMnYnBit k edIm,I[ a b c k .a b c .

3 3 3

3

3

3

dMeNaHRsay k¿sresr a b c nig a b c CaTRmg´BICKNit ½ eyIgman a (2 3i) 8 36i 54 27i 46 9i

3 3 3 3 3

b 3 (3 i) 3 27 27i 9 i 18 26i c 3 (1 4i) 3 1 12i 48 64i 47 52i

eyIgán a b c 46 9i 18 26i 47 52i 111 87i dUcen¼ a b c 111 87i . eyIgman a b c (2 3i)(3 i)(1 4i)

3 3 3

3

3

3

(6 2i 9i 3)(1 4i)

(9 7i)(1 4i) 9 36i 7i 28 37 29i a b c 37 29i

dUcen¼

.

9

x¿kMnt´cMnYnBit k eyIgman a b c k .a b c a b c eKTaj k a.b.c dUcen¼ k 3 .

3 3 3 3 3

3

111 87i 3 37 29i

lMhat;TI2 eK[cMnYnkMupøic a 3 i ; u (x 1) i.(y 2) nig Edl x nig y CaBIrcMnYnBit . cUrkMnt´témø x nig y edIm,I[ au b 0 .

b 2 16i

dMeNaHRsay kMnt´témøén x nig eyIgán au b 0

u

y

b a

eday a 3 i ; u (x 1) i.(y 2) nig eKán (x 1) i.(y 2) 2316i i

b 2 16i

10

eKTaján dUcen¼ x 2 ; y 3

2(1 8i)(3 i) 32 i 2 2(3 i 24i 8) ( x 1) i( y 2) 10 2(5 25i) ( x 1) i( y 2) 10 ( x 1) i( y 2) 1 5i x 1 1 x 2 ; y3 y25 ( x 1) i( y 2)

naM[ .

11

lMhat;TI3 eK[cMnYnkMupøic Z log ( x y ) i log x log y 2 nig W 132ii Edl x IR ; y IR . 1 k-cUrsresr W CaTRmg´BICKNit . x-kMnt´ x nig y edIm,I[ Z W .

3 2 2 * *

dMeNaHRsay k-sresr W CaTRmg´BICKNit )( eyIgán W 122ii (12 1i 14 2i) 12 245i i 2 2 5i 1 dUcen¼ W 2 5i . x-kMnt´ x nig y edIm,I[ Z W xy log ( eyIgán Z W smmUl 3 ) 2

2

log 2 x log 2 y 5

¦

xy )2 log 2 ( 3 log 2 ( x.y) 5

naM[

x y 12 x.y 32

2

eKTaj x ; y Ca¦ssmIkar u 12u 32 0 u eday ' 36 32 4 eKTaj¦s u 66 22 48 dUcen¼ x 4 ; y 8 ¦ x 8 ; y 4 .

1 2

12

lMhat;TI4 eK[smIkar (E): z az b 0 Edl a ; b IR cUrkMnt´témø a nig b edIm,I[cMnYnkMupøic z 2 i 3 Ca¦srbs´smIkar (E) rYcTajrk¦s z mYYyeTot rbs´smIkar . etIGñkBinitüeXIjdUcemþccMeBa¼cMnYnkMupøic z nig z ?

2

1

2

1

2

dMeNaHRsay kMnt´témø a nig b edIm,I[cMnYnkMupøic 2 i 3 Ca¦srbs´smIkarlu¼RtaEt vaepÞógpÞat´smIkar . eKán (2 i 3) a (2 i 3) b 0

2

4 4i 3 3 2a ai 3 b 0 (1 2a b) i(4 3 a 3 ) 0 3 eKTaján 41 aa 3b 00 ¦ ab 4 2 7 dUcen¼ a 4 ; b 7 . KNna¦smYYyeTotrbs´smIkar ½ ebI z ; z Ca¦srbs´smIkarena¼tamRTwsþIbTEvüteKán z z a 4 eday z 2 i 3

1 2 1 2

1

13

eKTaj z 4 (2 i 3) 2 i 3 dUcen¼ z 2 i 3 . eyIgBinitüeXIjfa z 2 i 3 nig CacMnYnkMupøicBIrqøas´Kña .

2 2 1

z2 2 i 3

lMhat;TI5 eK[cMnYnkMupøic z Edlman z CacMnYnkMupøicqøas´rbs´va eda¼RsaysmIkar log | z | z 7i z 3 i

5

dMeNaHRsay eda¼RsaysmIkar ½

log 5 | z | z iz 3i 7

tag z x i.y naM[ z x iy Edl smIkarGacsresr ½

log 5 | x iy | ( x iy) i( x iy) 3i 7 x iy ix y 3i log 5 x 2 y 2 7 xy xy log 5 x 2 y 2 ) i 3i ( 7 7

x ; y IR

14

eKTaján

¦ eday naM[ eKánRbB&næ eKTajcemøIy x 3 , y 4 ¦ x 4 , y 3 dUcen¼ z 3 4i ; z 4 3i CacemøIyrbs´smIkar .

1 2

x y 7 1 x y 7 xy 1 log 5 x 2 y 2 3 2 2 log 5 x y 3 7 x y 7 x 2 y 2 ( x y) 2 2 xy xy 12 2 2 x y 25 x y 7 3 4 x.y 12 3 4

¦

lMhat;TI6 eK[cMnYnkMupøicBIr ½ z 3x i (2 x y) nig W 1 y i [ 1 2 Edl x nig y CacMnYnBit . kMnt´ x nig y edIm,I[ W z . dMeNaHRsay kMnt´ x nig y edIm,I[ W z lu¼RtaEt

log 5 ( x 3 )

]

Re( W ) Re(z ) Im( W ) Im(z )

15

eKTaján ( 2) tam (1) eKTaj y 3x 1 (3) ykCYskñúgsmIkar (2) eKán 1 2 2x 3x 1 2 x lk&çx&NÐ x 3 0 ¦ x 3 tag t log (x 3) naM[ x 3 5 ¦ x 5 3 eKánsmIkar 2 5 3 ¦ 5 2 3 edayGg:xageqVgénsmIkarCaGnuKmn_ekIn nig Gg:xageqVg CaGnuKmn_efrena¼eyIgánsmIkarman¦sEtmYyKt´KW t 1 cMeBa¼ t 1 eKán x 5 3 2 nig y 3(2) 1 7 dUcen¼ x 2 ; y 7 .

log 5 ( x 3 ) log 5 ( x 3 ) t t 5 t t t t

1 y 3x log 5 ( x 3 ) 2 x y 1 2

(1)

16

lMhat;TI7 eda¼RsaysmIkar

2z | z |

9 7i 1 i

Edl

z

CacMnYnkMupøic .

dMeNaHRsay eda¼RsaysmIkar ½ 9 7i 2z | z | tag z x i.y , x; y IR 1 i i eKán 2(x iy) | x iy | (917i)()(11i)i) (

2 x 2iy x 2 y 2 9 9i 7i 7 2 (2 x x 2 y 2 ) 2iy 1 8i 2 x x 2 y 2 1 (1) 2 y 8 ( 2)

eKTaj tam (2) eKTaj

2 x x 2 16 1

y 4

ykeTACYskñúg (1) eKán

1 2

2 x 1 x 2 16 (2 x 1) 2 x 2 16 4 x 2 4 x 1 x 2 16 3x 2 4 x 15 0

, x

(minyk) eKTaj¦s eKán x 3 ; y 4 . dUcen¼ z 3 4i CacemøIyrbs´smIkar .

; ' 4 45 7 2 27 27 5 1 x1 3 ; x2 3 3 3 2

17

lMhat;TI8 eK[cMnYnkMupøicBIr Z nig Z Edl Z Z Z cUrRsaybBa¢ak;fa Z || Z || .

1 2 1 1 2 2

2

0

.

dMeNaHRsay Z Z RsaybBa¢ak;fa Z || Z || eyIgtag Z a i.b nig Z c i .d Edl a , b , c , d CacMnYnBit . Z b b)( d eyIg)an Z ac ii..d ((ac ii..d)(cc ii..d)) ac i.ad ii.bc i .bd Et i 1 c .d Z ad eK)an Z (ac bdc) id(bc ad) ac bd i. bc d c d c Z ad naM[ Z ac bd bc d c d c

1 1 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2

(ac bd) 2 ( bc ad) 2 c2 d 2 a 2 c 2 2abcd b 2 d 2 b 2 c 2 2abcd a 2 d 2 c2 d 2 (a 2 c 2 a 2 d 2 ) ( b 2 c 2 b 2 d 2 ) c2 d 2 a 2 ( c 2 d 2 ) b 2 (c 2 d 2 ) (c 2 d 2 )(a 2 b 2 ) c2 d 2 c2 d2 | Z1 | a 2 b 2 Z1 a 2 b2 Z2 c2 d 2 | Z2 | c2 d 2 Z1 |Z | 1 Z2 | Z2 |

eKTaj dUecñH

edayeKman .

18

lMhat;TI9 eK[cMnYnkMupøicBIr Z nig cUrRsaybBa¢ak;fa | Z Z

1 1

Z2

2

. .

| | Z1 | | Z 2 |

dMeNaHRsay RsaybBa¢ak;fa | Z Z | | Z || Z | eyIgtag Z a i.b nig Z c i .d Edl a , b , c , d CacMnYnBit . eyIgman Z Z (a i.b)(c i.d) ac i.ad i.bc i .bd Et i 1 naM[ Z Z (ac bd) i.(ad bc) eK)an | Z Z | (ac bd) (ad bc)

1 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2

a 2 c 2 2abcd b 2 d 2 a 2 d 2 2abcd b 2 c 2 (a 2 c 2 b 2 c 2 ) (a 2 d 2 b 2 d 2 ) c 2 (a 2 b 2 ) d 2 ( a 2 b 2 ) (a 2 b 2 )(c 2 d 2 )

eKTaj | Z Z | a b |Z | a b edayeKman | Z | c d

1 2

2 2 1 2 2 2

2

2

c2 d 2

dUcenH

| Z 1 Z 2 | | Z1 | | Z 2 |

.

19

lMhat;TI10 eK[cMnYnkMupøicBIr Z nig Z . k> cUrRsaybBa¢ak;fa | Z Z | | Z | | Z | x> Taj[)anfa (a c) (b d) a b cMeBaHRKb; a , b , c , d CacMnYnBit .

1 2 1 2 1 2

2 2 2 2

c2 d 2

dMeNaHRsay k> RsaybBa¢ak;fa | Z Z | | Z | | Z | kñúgbøg;kMupøic (XOY) eyIgeRCIserIsviucTr½BIr U nig V manGahVikerogKña Z nig Z naM[viucTr½ U V manGahVik Z Z . tamlkçN³RCugrbs;RtIekaNeK)an || U V || || U || || V || eday ³

1 2 1 2 1 2 1 2

|| U || | Z1 | , || V || | Z 2 |

|| U V ||| Z1 Z 2 |

dUcenH | Z Z | | Z | | Z | x> Taj[)anfa (a c) (b d) a b c d eyIgtag Z a i.b nig Z c i .d Edl a , b , c , d CacMnYnBit . man Z Z (a c) i.(b d) naM[ | Z Z | (a c) (b d)

1 2 1 2 2 2 2 2 2 2 1 2 2 2 1 2 1 2

20

ehIy | Z | a b , | Z | c d . tamsRmayxagelIeKman | Z Z | | Z | | Z | dUcenH (a c) (b d) a b c d

1 2 1 2 1 2 2 2 2 2 2 2

2

2

2

2

.

lMhat;TI11 eK[cMnYnkMupøic ³

Z1 3 4 i , Z 2 12 5 i , Z 3 8 15 i

k> cUrKNna | Z | , | Z | , | Z | nig | Z Z Z | x> Tajbgðajfa | Z Z Z | | Z | | Z | | Z nig | Z | | Z Z Z | | Z | | Z | . dMeNaHRsay k> KNna | Z | , | Z | , | Z | nig | Z Z Z | eyIg)an | Z | 3 4 9 16 25 5

1 2 3 1 2 3 1 2 3 1 2 1 1 2 3 2 3 1 2 3 1 2 3 2 2 1

3

|

| Z 2 | 12 2 5 2 144 25 169 13 | Z 3 | (8) 2 (15) 2 64 225 289 17

ehIy eK)an dUcenH

Z1 Z 2 Z 3 (3 4i) (12 5i) (8 15i) 7 24i | Z1 Z 2 Z 3 | 7 2 24 2 49 576 625 25 | Z1 | 5 , | Z 2 | 13 , | Z 3 | 17

nig | Z Z

1

2

Z 3 | 25

.

21

x> Tajbgðajfa ³

| Z 1 Z 2 Z 3 | | Z1 | | Z 2 | | Z 3 |

nig | Z | | Z Z Z | | Z | | Z | eyIgman | Z Z Z | 25 ehIy | Z | | Z | | Z | 5 13 17 35 dUcenH | Z Z Z | | Z | | Z | | Z | . mü:ageTot | Z | | Z Z Z | 5 25 30 nig | Z | | Z | 13 17 30 dUcenH | Z | | Z Z Z | | Z | | Z | .

1 1 2 3 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 2 3 2 3 1 1 2 3 2 3

lMhat;TI12 eK[smIkar (E) : z a z b 0 , a , b IR cUrkMnt;cMnYnBit a nig b edIm,I[cMnYnkMupøic z 3 2i Cab¤srbs; smIkarrYcKNnab¤smYyeTotrbs;smIkar . dMeNaHRsay kMnt;cMnYnBit a nig b ³ edIm,I[cMnYnkMupøic z 3 2i Cab¤srbs;smIkarlu³RtaEtvaepÞógpÞat;nwg smIkar .

2

22

eyIg)an (3 2i)

2

a (3 2i) b 0

9 12i 4i 2 3a 2ia b 0 , i 2 1

9 12i 4 3a 2ia b 0 (5 3a b) i (12 2a) 0

eKTaj)an 1232ab0 0 naM[ a 6 , b 13 5 a dUcenH a 6 , b 13 . KNnab¤smYyeTotrbs;smIkar ³ cMeBaH a 6 , b 13 smIkarkøayCa Z 6Z 13 0 tamRTwsþIbTEvüteyIgman Z Z 6 edayeKsÁal; Z 3 2i eK)an Z 6 Z 6 (3 2i) 3 2i dUcenHb¤smYyeToténsmIkarKW Z 3 2i . lMhat;TI13 eK[cMnYnkMupøic z 2 2 i. 2 2 . k>cUrsresr z CaTMrg;BiCKNit . x>cUrsresr z nig z CaTMrg;RtIekaNmaRt . K>TajrktMélR)akdén cos nig sin . 8 8

2

1

2

1

2

1

2

2

2

dMeNaHRsay k-sresr z CaTMrg;BiCKNit eyIg)an z ( 2 2 i. 2

2

23

2

2)

2

( 2 2 ) 2 2( 2 2 )(i. 2 2 ) (i. 2 2 ) 2 2 2 2 22 2.i 2 2 2 2 2 2i

dUcenH z 2 2 2 2.i x-cUrsresr z nig z CaTMrg;RtIekaNmaRt eKman z 2 2 2 2.i 4 22 i. 22 4 cos i.sin 4 4

2

2

2

dUcenH K-TajrktMélR)akdén tamsMrayxagelIeKTaj dUcenH

cos

z 2 4 cos i. sin 4 4 cos 8

nig nig

sin

z 2 cos i. sin 8 8 sin 8

2 cos i.sin 2 2 i. 2 2 8 8

2 2 8 2

nig

2 2 8 8

.

24

lMhat;TI14 eKeGaycMnYnkMupøic ³ z 6 2i. 2 nig z 1 i k>cUrsresr z , z nig Z zz CaragRtIekaNmaRt. x>cUrsresr Z zz CaragBiCKNit.

1 2 1 1 2 2 1 2

K>TajeGay)anfa cos 12 6 2 nig sin 12 6 2 . 4 4 dMeNaHRsay k>sresr z , z nig Z zz CaragRtIekaNmaRt³ eKman z 6 2i 2 2 23 i. 1 2 cos i. sin 2 6 6 dUcenH z 2 cos( ) i. sin( ) . 6 6 eKman z 1 i 2 22 i. 22 2 cos i. sin 4 4 dUcenH z 2 cos( ) i. sin( ) . 4 4 eKman Z zz 2 cos( ) i. sin( ) 6 4 6 4 2 dUcenH Z cos 12 i. sin 12 . x> sresr Z zz CaragBiCKNit eK)an Z 26(1i i)2 ( 26(1i i)(21)(1i) i) 6 i 6 i 2 2 4

1 1 2 2 1 1 2 2 1 2 1 2

25

dUcenH Z 6 2 i. 6 4 4 K> TajeGay)anfa ³ 6 2 cos nig sin 12 12 4 tamsRmayxagelIeKman ³

Z cos

2

.

6 2 4

i. sin (1) 12 12

nig Z 6 2 i. 6 2 (2) 4 4 pÞwmTMnak;TMng ¬!¦ nig ¬@¦ eK)an ³

cos

dUcenH

6 2 6 2 i. sin i. 12 12 4 4 6 2 6 2 cos sin 12 4 12 4

nig

.

26

lMhat;TI15 eK[cMnYnkMupøic z 2 2 3 i. 2 2 3 k-cUrbgðajfa z 2(cos 24 i.sin 24 ) . x-cUrsresr Z 2z z CaTRmg;RtIekaNmaRt . K-cUrKNnatémøén S z z CaGnuKmn_én n .

2 n n n

dMeNaHRsay k-bgðajfa z 2(cos 24 i.sin 24 ) eKman z 2 2 3 i. 2 2 3 tag a 2 2 3 nig b 2 2 eyIg)an a 2 2 2. 23

2 2 2cos 2 4cos2

3

2 2(1 cos ) 6 6

2 2cos 4cos2 2cos 12 12 24 24

mü:ageTot b eK)an dUcenH

2 2 3

4 sin 2 2 sin 12 24 24 z 2 cos 2i. sin 2(cos i. sin ) 24 24 24 24 z 2 cos i. sin 24 24 2 2 cos

.

27

x-sresr CaTRmg;RtIekaNmaRt ³ eyIgman z 2(cos 24 i.sin 24 ) eyIg)an

i. sin )]2 24 24 Z 2 2(cos i. sin ) 24 24 4(cos i.sin ) 2(cos i.sin ) 12 12 12 12 2(1 cos i.sin ) 2 cos 2 2i.sin . cos 24 24 48 48 48 2(cos i.sin ) 12 12 2 cos (cos i.sin ) 48 48 48 1 cos( ) i.sin( ) 12 48 12 48 cos 48 1 3 3 1 cos i.sin cos i.sin 48 48 cos 16 16 cos 48 48 1 Z cos i. sin 16 16 cos 48

[2(cos

z2 Z 2z

dUcenH

.

K-KNnatémøén S z z tamrUbmnþdWmr½eyIgman ³ n n n n z 2 cos i.sin nig z 2 cos i. sin 24 24 24 24 eyIg)an S 2 cos n i.sin n 2 cos n i.sin n 24 24 24 24 dUcenH S 2 .cos n . 24

n n n n n n n n n 1 n

n

n

28

lMhat;TI16 eK[cMnYnkMupøic Z cos 45 i.sin 45 cUrRsaybBa¢ak;fa (1 Z) 8 cos 25 (cos 65 i.sin 65 ) .

3 3

dMeNaHRsay RsaybBa¢ak;fa ³

(1 Z) 3 8 cos 3

eyIgman tamrUbmnþ eyIg)an b¤ tamrUbmnþdWmr½eyIg)an ³

2 6 6 (cos i.sin ) 5 5 5 4 4 1 Z 1 cos i.sin 5 5 1 cos 2 cos 2 sin 2 sin cos 2 2 2 2 2 2 1 Z 2 cos 2 2i sin cos 5 5 5 2 2 2 1 Z 2 cos (cos i.sin ) 5 5 5

nig

dUcenH

2 2 2 i.sin ) (1 Z) 3 2 cos (cos 5 5 5 2 6 6 (cos i.sin ) 8 cos 3 5 5 5 2 6 6 (1 Z) 3 8 cos 3 (cos i.sin ) 5 5 5

3

.

29

lMhat;TI17 eK[cMnYnkMupøic Z 4 2 (1 i) k-cUrsresr Z CaTRmg;RtIekaNmaRt . x-KNnab¤sTI# én Z . dMeNaHRsay k-sresr Z CaTRmg;RtIekaNmaRt ³ eyIg)an Z 4 2 (1 i)

2 2 i. ) 8( cos i.sin ) 2 2 4 4 3 3 8cos( ) i.sin( ) 8(cos i.sin ) 4 4 4 4 3 3 Z 8 (cos i.sin ) 4 4 8 (

. dUcenH x-KNnab¤sTI# én Z eyIgtag W Cab¤sTI# én Z 8 (cos 34 i.sin 34 ) tamrUbmnþb¤sTI n : W r cos( n2k ) i.sin( n2k )

k n k

eyIg)an -ebI -ebI -ebI

3 3 2k 2 k Wk 3 8 cos ( 4 ) i.sin( 4 ) 3 3 k 0 : W0 2 (cos i.sin ) 4 4 11 11 i sin ) k 1 : W1 2(cos 12 12 19 19 i.sin ) k 2 : W2 2 (cos 12 12

.

30

lMhat;TI18 cUrKNnab¤sTI n éncMnYnkMupøic Z 1 cMeBaH n 2 , 3 , 4 , 5 , 6 . dMeNaHRsay KNnab¤sTI éncMnYnkMupøic Z 1 cMeBaH n 2 , 3 , 4 , 5 , 6 eyIgman Z 1 cos( 2k) i.sin( 2k) tag W Cab¤sTI n én Z eyIg)an ³

n

k

Wk cos

k> cMeBaH eK)an -ebI -ebI x> cMeBaH eK)an -ebI -ebI k 1 : W cos i.sin 1 -ebI k 2 : W cos 53 i sin 53 1 i. 23 2 K> cMeBaH n 4 eK)an W cos (2k 4 1) i.sin (2k 4 1) -ebI k 0 : W cos i.sin 22 i. 22 4 4

1 2 k 0

(1 2k ) (1 2k ) i.sin n n (1 2k ) (1 2k ) i.sin Wk cos n2 2 2 k 0 : W0 cos i.sin i 2 2 3 3 k 1 : W1 cos i.sin i 2 2 (2k 1) (2k 1) Wk cos i.sin n3 3 3 1 3 k 0 : W0 cos i.sin i. 3 3 2 2

31

-ebI k 1 : W cos 34 i.sin 34 22 i 22 -ebI k 2 : W cos 54 i.sin 54 22 i 22 -ebI k 3 : W cos 74 i.sin 74 22 i. 22 X> cMeBaH n 5 eK)an W cos (1 52k) i.sin (1 52k) -ebI k 0 : W cos i.sin 5 5 -ebI k 1 : W cos 35 i.sin 35 -ebI k 2 : W cos i.sin 1 -ebI k 3 : W cos 75 i.sin 75 -ebI k 4 : W cos 95 i.sin 95 g> cMeBaH n 6 eK)an W cos (1 62k) i.sin (1 62k) -ebI k 0 : W cos i sin 23 i 1 6 6 2 -ebI k 1 : W cos i.sin i 2 2 -ebI k 2 : W cos 56 i sin 56 23 i 1 2 -ebI k 3 : W cos 76 i.sin 76 23 i 1 2 -ebI k 4 : W cos 32 i.sin 32 i -ebI k 5 : W cos 11 i.sin 11 23 i 1 . 6 6 2

1 2 3 k 0 1 2 3 4 k 0 1 2 3 4 5

32

lMhat;TI19 KNnab¤sTI n éncMnYnkMupøic cMeBaH n 2 , 3 , 4 , 5 , 6 .

Zi

dMeNaHRsay KNnab¤sTI n éncMnYnkMupøic Z i cMeBaH n 2 , 3 , 4 , 5 , 6 eyIgman³ Z i cos( 2k) i.sin( 2k) 2 2 tag W Cab¤sTI n én Z i cos( 2k) i.sin( 2k) 2 2 tamrUbmnþeK)an W cos (1 24nk) i.sin (1 24nk) k> cMeBaH n 2 eK)an W cos (1 44k) i.sin (1 44k) -ebI k 0 : W cos i.sin 22 i 22 4 4 -ebI k 1 : W cos 54 i.sin 54 22 i 22 x> cMeBaH n 3 eK)an W cos (4k 6 1) i.sin (4k 6 1) -ebI k 0 : W cos i.sin . 23 i 1 6 6 2 -ebI k 1 : W cos 56 i.sin 56 23 i 1 2

k k k 0 1 k 0 1

33

-ebI k 2 : W cos 32 i sin 32 i K> cMeBaH n 4 eK)an W cos (4k 8 1) i.sin (4k 8 1) -ebI k 0 : W cos i.sin 8 8 -ebI k 1 : W cos 58 i.sin 58 -ebI k 2 : W cos 98 i.sin 98 -ebI k 3 : W cos13 i.sin 13 8 8 4 4 X> cMeBaH n 5 eK)an W cos (1 10k) i.sin (1 10k) -ebI k 0 W cos 10 i.sin 10 -ebI k 1 W cos i.sin i 2 2 9 9 -ebI k 2 W cos 10 i.sin 10 -ebI k 3 W cos13 i.sin 13 10 10 -ebI k 4 W cos 17 i.sin 17 10 10 4 4 g> cMeBaH n 6 eK)an W cos (1 12k) i.sin (1 12k) -ebI k 0 W cos 12 i sin 12 5 5 -ebI k 1 W cos 12 i.sin 12 -ebI k 2 W cos 34 i sin 34 22 i 22 -ebI k 3 W cos13 i.sin 13 12 12

2 k 0 1 2 3 k 0 1 2 3 4 k 0 1 2 3

34

-ebI k 4 -ebI k 5

W4 cos

17 17 i.sin 12 12 7 7 2 2 W5 cos i.sin i 4 4 2 2

.

35

lMhat;Gnuvtþn_

!> cUrsresrcMnYnkMupøicxageRkamCaTRmg;BICKNit Z a i.b ³ k> a / Z (2 5i)(3 i) b / Z (1 i)(1 2i)(1 3i) x> a / Z ( 2 3 i ) (1 2 i ) b / Z (3 2i) (1 2i) b / Z (1 2i) ( 2 i) K> a / Z (1 2i) (1 i) 9 7i X> a / Z 427ii b/Z 3 2i 1 2 i b/Z g> a / Z (2 i1)(22i 3i) 1 2 i @> cUrkMnt;BIrcMnYnBit a nig b edIm,I[cMnYnkMupøic z 2 3i Cab¤smYyrbs;smIkar (E): z az b 0 . #> cUrkMnt;BIrcMnYnBit a nig b edIm,I[cMnYnkMupøic z 1 2i Cab¤smYyrbs;smIkar (E): z az b 0 . $> eKmansmIkar (E): z 3z 4 6i 0 k-kMnt;cMnYnBit b edIm,I[ z b.i Cab¤srbs;smIkar (E) . x-cUredaHRsaysmIkar (E) kñúgsMNMukMupøic . %> eKmansmIkar (E): z 5(1 i)z 3(4 3i) 0 k-cUrepÞógpÞat;fa 48 14i (7 i) x-cUredaHRsaysmIkar (E) kñúgsMNMukMupøic .

2 2

2 2 3 3 3 3 2 2 3 2 0 2 2

36

^> eKmansmIkar (E): z (a i.b)z a 3 5i 0 Edl a , b IR . kMnt;témø a nig b edIm,I[ z 3 2i Cab¤smYyrbs;smIkar (E) rYccUrkMnt;rkb¤s z mYyeTotcMeBaHtémø a nig b Edl)anrkeXIj . &> eKmansmIkar (E) :z (5 i)z (10 9i)z 2(1 8i) 0 . k-kMnt;cMnYnBit b edIm,I[ z b.i Cab¤srbs;smIkar (E) . x-cUrsresrsmIkar (E) Carag (z z )(z pz q) 0 Edl p nig q CacMnYnkMupøiucBIrEdleKRtUvrk . K-cUrepÞógpÞat;fa 8 6i (1 3i) rYcedaHRsay smIkar (E) kñúgsMNMukMupøic. *> cUredaHRsaysmIkarxageRkamkñúgsMNMukMupøic ³

1 2 3 2 0 2 0 2

2

a / iz 2 (2 3i) z (1 5i) 0 b / ( 2 i) z 2 5(1 i) z 2(3 4i) 0 c / (1 i)z 2 (1 7i)z 2( 2 3i) 0

(> cUrbegáItsmIkardWeRkTIBIrEdlman nig Cab¤skñúgkrNInImYy² xageRkam ³ k> a / 2 11i , 2 11i b / 2 3i , 2 3i x> a / 3 2i , 1 3i b / 2 3i , 3 2i K> a / 1 2i , 3 2i b / 3 i , 1 i 3 !0> eKmancMnYnkMupøic 1 3i nig 1 2i . cUrsresrsmIkardWeRkTIBIrmYyEdlman Z

2 2

37

1

nig Z . Cab¤s> !!> cUrkMnt;rkcMnYnkMupøicEdlmanm:UDúlesµI 8 ehIykaerrbs;va CacMnYnnimµitsuTæ. [email protected]> cUrkMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ 8 9i (1 i)( x iy) (3 2i)( x iy) . 1 2i !#> cUrkMnt;BIrcMnYnBit x nig y ebIeKdwgfa ³ 5 13i (3 2i) ( x y) (1 2i)( x y) . 1 i !$> eKmansmIkar (E) : az bz c 0 , a 0 , a, b, c IR . cUrbgðajfaebI z Cab¤ssmIkar (E) enaH z k¾Cab¤srbs; (E) Edr . i !%> eKmancMnYnkMupøic 1 2i 5 nig 1 2 5 . k-cUrbegáItsmIkardWeRkTIBIrmYyEdlman nig Cab¤s . x-eKtag S cMeBaHRKb; n IN . cUrRsaybBa¢ak;TMnak;TMng S S 3 S 0 ? 2 i K-edaymin)ac;BnøatcUrKNna N 1 2i 5 1 2 5 . !^> eKmansmIkar (E) : z ( )iz 0 , , IR * cUrbgðajfasmIkar (E) b¤sBIrsuTæEtCacMnYnnimµitsuTæEdleKnwgbBa¢ak; !&> eKmansmIkardWeRkTIBIr (E) : Az Bz C 0 Edl A 0 ehIy A , B, C CacMnYnkMupøic .

2 2 0 0 n n n n 2 n 1 n 10 10 2 2

2

2

38

k> bgðajfaebI A C i.B enaHsmIkar (E) manb¤sBIrkMnt;eday ³ C z i , z i. . A x> bgðajfaebI A C i.B enaHsmIkar (E) manb¤sBIrkMnt;eday ³ C z i , z i. . A K> Gnuvtþn_ ³ cUredaHRsaysmIkarxageRkamkñúgsMNMukMupøic ³

1 2 1 2

a / (1 i) z 2 (2 3i)z 2 3i 0 b / (1 2i)z 2 (1 2i)z (1 i) 0

!*> eK[cMnYnkMupøic Z 1 2i nig Z 3 i . cUrkMnt;rkEpñkBit nigEpñknimµiténcMnYnkMupøic W 1 ebIeKdwgfa W Z1 Z1 . !(> eKmancMnYnkMupøic nig Edl 3 2i nig . 5(1 i) . cUrkMnt;EpñkBit nig Epñknimµitén Z 1 1 . @0> eKmansmIkar (E): i.Z (2 3i) Z 5(1 i) 0 k> kMnt;cMnYnBit a edIm,I[ Z a i Cab¤smYyrbs;smIkar (E) rYcKNnab¤s Z mYyeTotrbs;smIkar . q x> cUrkMnt;cMnYnBit p nig q edIm,I[ Zp Zq pZ Z 2 . @!> eKmancMnYnkMupøicBIr Z nig Z Edl Z Z 3 i nig Z .Z 4 3i k> cUrbgðajfa (Z ) (Z ) 0 . x> cUrkMnt;rkEpñkBit nig Epñknimµit én Z Z Z .

1 2 1 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 1 2

39

4

4

1

2

K> cUrepÞógpÞat;fa Z Z 4Z .Z (Z Z ) rYckMnt;rk Z nig Z . @@> eKmancMnYnkMupøic Z x i xy nig Z y ixy Edl x, y IR . cUrkMnt; x nig y edIm,I[ Z Z ( 2006 i 2007 ) . @#> eK[cMnYnkMupøic ³ Z (a b) i(b c) , Z (b c) i(c a ) nig Z (c a ) i(a b) Edl a , b , c CabIcMnYnBitxusKña. cUrbgðajfa Z Z Z 3Z .Z .Z @$> cUrKNnab¤skaeréncMnYnkMupøicxageRkam ³ k> a / Z 48 14i b / Z 24 10i x> a / Z 40 42i b / Z 77 36i K> a / Z 55 48i b / Z 1 i 6 @%> cUrsresrcMnYnkMupøicxageRkamCaTRmg;RtIekaNmaRt ³ k> a / Z 1 i 3 b / Z 2 3 2i x> a / Z 1 2i b / Z 3 3i K> a / Z 6 2i 2 b / Z 2 2i X> a / Z 1 i3 b/Z 2 i 2 g> a / Z 1 2 i b/Z 2 3 i

1 2 1 2 1 2 1 2 2 2 1 2 2 1 2 1 2 2 3 3 3 1 2 3 1 2 3

2

2

40

@^> cUrsresrCaTRmg;RtIekaNmaRténcMnYnkMupøicxageRkam ³ k> a / Z 1 cos 49 i.sin 49 b / Z 1 sin i. cos 10 10 4 4 x> a / Z sin 27 i(1 cos 27 ) b / Z 1 cos i sin 5 5 K> a / Z 2 2 cos i. 2 2 cos b / Z 1 i tan 7 5 5 b / Z ( 3 2)(cos i sin ) X> a / Z sin 10 i.cos10 8 8 g> Z 2 2 2 2 cos i. 2 2 2 2 cos . 4 4 @&> cUredaHRsaysmIkarxageRkamrYcsresrb¤snImYy² CaragRtIekaNmaRt ³ k> a / z 2z 4 0 b / 2z 2 z 1 0 x> a / z 3.z 1 0 b / z z 1 0 @*> cUrsresr Z 2 3 i. 2 3 CaragRtIekaNmaRt . @(> cUrsresr Z 2 2 i 2 2 CaragRtIekaNmaRt . #0> k-KNnatémøR)akdén sin 10 , cos10 nig tan 10 . x-TajrkTRmg;RtIekaNmaRtén Z 1 i. 5 2 2 . #!> eK[cMnYnkMupøic z 2 2i 6 nig z 1 i k> cUrsresr Z zz CaTRmg; a b.i . x> cUrsresr z , z nig Z zz CaTRmg;RtIekaNmaRt .

2

2

2

2

1

2

1

2

1

1

2

2

41

K> edayeRbIlTæplxagelIcUrTajrktémøR)akdén cos 12 nig sin 12 . #@> eKmancMnYnkMupøic z 1 2i nig z 2 3 2i . k> cUrsresr Z z .z CaTRmg;BICKNit . x> cUrsresr z , z nig Z z .z CaTRmg;RtIekaNmaRt . 5 5 K> edayeRbIlTæplxagelIcUrTajrktémøR)akdén cos 12 nig sin 12 . ##> eK[ z 1 i . cUrsresr z nig z CaTRmg;RtIekaNmaRt . #$> eK[ z 3 i . cUrsresr z nig z CaTRmg;RtIekaNmaRt . #%> eK[ z 1 i 23 . cUrsresr z nig z CaTRmg;RtIekaNmaRt . 2 #^> eK[cMnYnkMupøic Z 2 3 i k-cUrepÞógpÞat;fam:UDúl | Z | 6 2 . x-bgðajfa Z 2(1 cos i.sin ) rYcTajrkTRmg;RtIekaNmaRt 6 6 én Z . K-Tajbgðajfa cos 12 6 2 . 4

1 2 1 2 1 2 1 2 2007 2007 2007

#&> eK[cMnYnkMupøic Z 2 1 i k-cUrepÞógpÞat;fam:UDúl | Z | 2. 2 2 . x-bgðajfa Z 2(1 cos i.sin ) rYcTajTRmg;RtIekaNmaRtén Z 4 4 K-Tajbgðajfa cos 2 2 2 . 8

42

#*> eKmancMnYnkMupøic ³ 3i Z nig Z ( 3 1) i( 3 1) . 2 k> cUrsresr U Z .Z CaragBICKNit . x> cUrsresr U nig Z CaragRtIekaNmaRt rYcTajrkm:UDúl nigGaKuym:g;éncMnYnkMupøic Z . K>cUrTajbgðajfa cos 12 6 2 nig sin 12 6 2 . 4 4 #(> eK[cMnYnkMupøic z 1 2i 3 . cUrrkm:UDúl nigGaKuym:g;éncMnYnkMupøic U 1z z . $0> eK[cMnYnkMupøic Z (nn nn11 i((22nn11)) ) Edl n CacMnYnKt;FmµCati . k> bgðajfa Z n 1 i n 11 i . x> KNna S Z Z Z ..... Z edaysresr lTæplEdl)anCaragBICKNit . $!> KmansIVúténcMnYnkMupøic (Z ) kMnt;eday ³ 3i 3 i i Z nig Z 1 2 3 Z 1 2 3 2 Edl n IN . k> eKtag U Z 1 cMeBaHRKb; n IN .

1 2 1 2 1 2 2007 2

2

n

2

2

2

n

n

0

1

2

n

n

0

n 1

n

n

n

43

i cUrbgðajfa U 1 2 3 .U , n IN x> cUrsresr U CaTRmg;RtIekaNmaRt . K> cUrRsaybBa¢ak;fa ³

n 1 n n

Z n 2 cos

(n 1) (n 1) (n 1) cos i. sin 6 6 6

n

X> cUrkMnt;TRmg;RtIekaNmaRtén Z . [email protected]> ebI z CacMnYnkMupøicEdlepÞógpÞat;TMnak;TMng ³ z (1 z) cMeBaHRKb; n IN enaHcUrRsaybBa¢ak; facMnYnkMupøic z nig 1 1 manm:UDúlesµIKña . z $#> edaHRsaysmIkar z z i | z | 1 i . i i $$> eKmancMnYnkMupøic 1 2 7 nig 1 2 7 k> sresrsmIkardWeRkTIBIrEdlman , Cab¤s . x> tag n Z : S . cUrRsayfa S S 2S 0 ?

2n n 3 3 2

n n n

n2

n 1

n

$%> eK[cMnYnkMupøic ³

1 2 3 1

Z1 a 1 i.b1 Z 2 a 2 i.b 2 Z a i.b 3 3 3

2 3

Edl a , a , a , b , b , b CacMnYnBit . kñúglMhRbkbedaytMruyGrtUnrm:al; (o, i , j, k)

44

eK[RtIekaN ABC mYyEdl AB(a , a , a ) nig AC(b , b , b ) . k>cUrkMnt;RbePTénRtIekaN ABC kalNaeKman TMnak;TMng Z Z Z 0 . x> cUrRsayfaebI ABC CaRtIekaNEkgsm)aTkMBUl A enaHeK)an (Z .Z .Z ) Z Z3 Z . $^> eK[cMnYnBit x Edl x k , k Z 2 k> cUrRsaybBa¢ak;fa ³

1 2 3 1 2 3

2 1 2 2 2 3

2

6 1

6 2

6 3

1

2

3

1 3 tan 2 x i.(3 tan x tan 3 x )

cos 3x i sin 3x cos 3 x

x>eRbITMnak;TMngxagelIcUrTajbgðajfa ³ K> cUrsresr nig tan( x) CaGnuKmn_ 3 én tan x . X> Taj[)anfa tan( x ) tan( x ) tan 3xx . 3 3 tan g>cUrKNnaplKuN P [tan( 3 a ). tan( 3 a )] . 3 3 $&-edaHRsaysmIkar | z | i.z 1 3i Edl z CacMnYnkMupøic . $*-edaHRsaysmIkar log (z.z) z 3 (1 2i) .

n n n n k 0 5

3 tan x tan 3 x tan 3x 1 3 tan 2 x tan( x ) 3

45

$(-edaHRsaysmIkar | z 1 i | iz 22 4i . %0-eK[sIVúténcMnYnkMupøic (Z ) kMnt;eday ³ 1 Z 1 nig Z ( Z | Z | ) Edl n IN 2 KNna Z edaysresrlTæpleRkamTRmg;RtIekaNmaRt .

n 0 n 1 n n

n

46

emeronTI2

lImIténGnuKmn_

1-niymn&y GnuKmn_ f (x) manlImItesµI L kalNa x xitCit x mann&yfacMeBa¼RKb´cMnYn 0 eKmancMnYn 0 Edl 0 | x x | eKán | f ( x ) L | . eKkMnt´sresr ½ lim f (x) L . 2-RTwsþIbTlImIt

0 0 xx0

1 / lim f ( x ) g ( x ) lim f ( x ) lim g ( x )

xx0 xx 0 xx 0

2 / lim k.f ( x ) k lim f ( x ) ; k IR

xx0 xx0

3 / lim f ( x ) . g ( x ) lim f ( x ) lim g ( x )

xx0 xx 0 xx0

lim f ( x ) x x 0 f ( x ) 4 / lim ; lim g ( x ) 0 xx 0 g( x ) lim g ( x ) x x 0 xx0

; n IN * 3-RTwsþIbTlImItGnuKmn_RtIekaNmaRt

xx0 xx0

5 / lim f ( x ) lim f ( x )

n

n

1 / lim

sin x x lim 1 x 0 sin x x 0 x tan x x 2 / lim lim 1 x 0 x 0 tan x x

47

4-lImIteqVg-sþaM ebIeKman lim f (x) L & lim f (x) L ena¼ lim f (x) L Edl lim f (x) & lim f (x) tagerogKñaCalImIteqVgnigsþaM 5-PaBCab´énGnuKmn_ k-PaBCab´Rtg´cMnucmYy GnuKmn_ y f (x) Cab´Rtg´cMnuc x x lu¼RtaEt f bMeBjl&kçx&NÐbIdUcxageRkam ½ 1¿ f (x ) mann&ylImIt ( f (a) CacMnYnBit ) 2¿ lim f (x) man ( lImItCacMnYnBit ) 3¿ lim f (x) f (x ) ( lImItesµInwgtémøénGnuKmn_Rtg´cMnuc x ) x-PaBCab´elIcenøa¼mYy -GnuKmn_ f (x) elIcenøa¼ebIk ] a; b [ lu¼RtaEt f ( x ) Cab´cMeBa¼RKb´cMnYnenAkñúgcenøa¼ebIkena¼ . -GnuKmn_ f (x) elIcenøa¼ebIk a ; b lu¼RtaEt f (x) Cab´elI ] a ; b [ nigman lim f (x) f (a) ; lim f (x) f (b) .

xx0 xx0 xx0 xx0 x x 0 0 0 x x 0 xx0 0 0 x a x b

48

6-RTwsþIbTtémøkNþal ebIGnuKmn_ f (x) Cab´elIcenøa¼ a ; b ehIy f (a).f (b) 0 ena¼y¨agehacNas´mancMnYn mYyenAcenøa¼ a nig b Edl f (x ) 0 .

0

y

y

x0

1

1

a

0 1

b

x

0 1

a

b

x

7-lImItRtg´Gnnþ ebI f (x) CaGnuKmn_mYy nig L CacMnYnBit . lim f ( x ) L mann&yfacMeBa¼RKb´ 0 eKmancMnYn M 0 Edl | f ( x ) L | kalNa x M . . lim f ( x ) L mann&yfacMeBa¼RKb´ 0 eKmancMnYn N 0 Edl | f ( x ) L | kalNa x N .

x x

49

8-lkçN¼lImItRtg´Gnnþ

1/ lim k f (x ) k lim f ( x ) ; 2 / lim f (x ) g( x ) lim f (x ) lim g( x )

x x x x x

3 / lim f (x ) g(x ) lim f ( x ) lim g( x )

x x x

4 / lim f (x ).g( x ) lim f (x ) lim g( x )

x x x

f (x ) xlim f (x ) 5 / lim ; lim g( x ) 0 x g ( x ) lim g( x) x x

(lkçN¼lImItRtg´ dUcKñanwglImItRtg´ Edr ) 9-GasIumtUténExßekag k¿ GasIumtUtQr GnuKmn_ y f (x) ebI lim f (x) ¦ lim f (x) ena¼eKfabnÞat´mansmIkar x a CaGasIumtUtQr énRkabtag f (x) .

x a x a

y

(C) : y = f(x)

1

x 0 1

x=a

x¿ GasIumtUtedk GnuKmn_ y f (x) ebI lim f (x) b ¦ lim f (x) b ena¼eKfabnÞat´mansmIkar y b CaGasIumtUtedk

x x

50

énRkabtag f (x) .

y

(C) : y = f(x) y = b

1

x 0 1

K¿ GasIumtUteRTt GnuKmn_ y f (x) ebIeKGacsresr f (x) ax b g(x) EdllImIt lim g(x) 0 ena¼eKfabnÞat´mansmIkar y ax b CaGasIumtUteRTténRkabtag f ( x ) .

x

y

(C) : y = f(x)

y = ax + b

1

x 0 1

51

lMhat;TI1 cUrKNnalImItxageRkam ³ x k> lim x xx 1 3 x> lim x x x x....... x 1 dMeNaHRsay KNnalImItxageRkam ³ k> lim x xx 1x 3

2 3 x 1

2 3 x 1 2 3 x 1

n

3

, n IN *

( x 1) ( x 2 1) ( x 3 1) lim x 1 x 1 ( x 1) ( x 1)( x 1) ( x 1)( x 2 x 1) lim x 1 ( x 1) lim1 ( x 1) ( x 2 x 1) 1 2 3 6 x 1 x x2 x3 3 6 lim x 1 x 1 x x 2 x 3 ....... x n 3 lim , n IN * x 1 x 1 (x 1) (x 2 1) (x 3 1) ..... (x n 1) lim x 1 x 1 (x 1) (x 1)(x 1) (x 1)(x 2 x 1) ... (x 1)(x n 1 ... x 1) lim x 1 x 1 2 n 1 lim1 (x 1) (x x 1) ... (x ... x 1) x 1

dUcenH > x>

.

1 2 3 ....... n

dUcenH

n(n 1) 2 x x 2 x 3 ...... x n n n (n 1) lim x 1 x 1 2

.

52

lMhat;TI2 cUrKNnalImItxageRkam ³ k> lim 1 1 x 1 3x x> lim(1 1 x 1 nx ) , n IN * K> lim 1 mx 1 nx , m, n IN * dMeNaHRsay KNnalImItxageRkam ³ k> lim 1 1 x 1 3x

x 1 3 x 1 n x 1 m n x 1 3

1 3 lim 2 x 1 1 x (1 x )(1 x x ) (1 x x 2 ) 3 lim 2 x 1 (1 x )(1 x x ) ( x 1) ( x 2 1) ( x 1) ( x 1)( x 1) lim lim x 1 (1 x )(1 x x 2 ) x 1 ( x 1)( x 2 x 1) 1 ( x 1) 1 2 lim 1 x 1 ( x 2 x 1) 3 3 1 lim 1 x 1 1 x 1 x3

dUcenH x> lim(1 1 x 1 nx ) , n IN *

x 1 n

.

53

1 n lim n 1 x 1 1 x (1 x )(1 x ... x ) (1 x ... x n 1 ) n lim x 1 (1 x )(1 x .... x n 1 ) ( x 1) .... ( x n 1 1) lim x 1 ( x 1)(1 x ... x n 1 ) ( x 1) ...... ( x 1)( x n 2 .... x 1) lim x 1 ( x 1)(1 x ..... x n 1 ) 1 ..... ( x n 2 .... x 1) 1 2 ..... (n 1) n 1 lim x 1 2 (1 x ..... x n 1 ) n 1 n n 1 lim( ) x 1 1 x 1 xn 2 n m lim , m, n IN * x 1 1 xm 1 xn n 1 m 1 lim 2 x 1 1 x 1 xm 1 x 1 x n 1 m 1 lim x 1 1 x 1 x n 1 x 1 x m

dUcenH K>

.

dUcenH

n m 1 1 lim lim n m x 1 1 x 1 x x 1 1 x 1 x n 1 m 1 n 1 m 1 m n 2 2 2 2 n mn m lim m x 1 1 x 1 xn 2

.

54

lMhat;TI3 cUrKNnalImItxageRkam ³ k> lim 1 (1 x)(1 x 2x)(1 3x) x> lim 1 (1 x)(1 2x)(x1 3x)......(1 nx) , n IN * dMeNaHRsay KNnalImItxageRkam k> lim 1 (1 x)(1 x 2x)(1 3x)

x 0 x 0 x 0

lim

1 (1 x ) (1 x )[1 (1 2 x )] (1 x )(1 2 x )[1 (1 3x )] x 0 x x 2 x (1 x ) 3x (1 x )(1 2 x ) lim x0 x lim1 2(1 x ) 3(1 x )(1 2 x ) (1 2 3) 6 x 0

x 0

dUcenH lim 1 (1 x)(1 x 2x)(1 3x) 6 . x> lim 1 (1 x)(1 2x)(x1 3x)......(1 nx) , n IN *

x 0

lim

1 (1 x ) (1 x )[1 (1 2x )] ... (1 x )(1 2x )...(1 (n 1) x )[1 (1 nx)] x 0 x x 2x (1 x ) ..... nx(1 x )(1 2x )....(1 (n 1) x ) lim x 0 x lim1 2(1 x ) 3(1 x )(1 2x ) .... n (1 x )(1 2x )...(1 (n 1) x

x 0

(1 2 3 ..... n )

n (n 1) 2

dUcenH

lim

1 (1 x )(1 2 x )(1 3x )....(1 nx ) n (n 1) x 1 x 2

.

55

lMhat;TI4 cUrKNnalImItxageRkam ³ k> lim (1 x) x (3x 1) x> lim (1 x) x (nx 1) , n IN * dMeNaHRsay KNnalImItxageRkam ³ k> lim (1 x) x (3x 1)

3 x 0 2 n x 0 2 3 x 0 2

dUcenH x> tamrUbmnþeTVFajÚtun (1 x)

1 nx lim

x0

1 3x 3x 2 x 3 3x 1 lim x 0 x2 3x 2 x 3 lim lim 3 x 3 2 x 0 x 0 x (1 x ) 3 (3x 1) 3 lim 2 x 0 x n (1 x ) (nx 1) , n IN * lim x 0 x2

n

.

2 n C 0 C1n x C n x 2 C 3 x 3 ... C n x n n n

n ( n 1) 2 n x C 3 x 3 .... C n x n nx 1 n 2 x2

n ( n 1) 2 x C 3 x 3 .... C n x n n n 2 lim x0 x2 n (n 1) n (n 1) n lim C 3 x .... C n x n 2 n x0 2 2

dUcenH

(1 x ) n (nx 1) n (n 1) lim 2 x 0 x 2

.

56

lMhat;TI5 cUrKNnalImItxageRkam ³ x k> lim 3x (x41) 1 x> lim nx (x(n)1)x 1 , n IN * 1 dMeNaHRsay KNnalImItxageRkam ³ x k> lim 3x (x41) 1

4 3 x 1 2 n 1 n x 1 2 4 3 x 1 2

(3x 4 3x 3 ) ( x 3 1) 3x 3 ( x 1) ( x 1)( x 2 x 1) lim lim x 1 x 1 ( x 1) 2 ( x 1) 2 3x 3 x 2 x 1 ( x 3 x 2 ) ( x 3 x ) ( x 3 1) lim lim x 1 x 1 x 1 x 1 x 2 ( x 1) x ( x 1)( x 1) ( x 1)( x 2 x 1) lim x 1 x 1 limx 2 x ( x 1) ( x 2 x 1) 1 2 3 6 x 1

dUcenH x>

3x 4 4 x 3 1 lim 6 2 x 1 ( x 1) nx n 1 (n 1) x n 1 lim , n IN * x 1 ( x 1) 2 nx n ( x 1) ( x 1)( x n 1 .... x 2 x 1) lim x 1 ( x 1) 2

.

( x n x n 1 ) .... ( x n x ) ( x n 1) lim x 1 x 1 n 1 x ( x 1) ... x ( x 1)( x n 2 ... x 1) ( x 1)( x n 1 ... x 1) lim x 1 x 1 n 1 n2 limx .... x ( x ... x 1) ...( x n 1 ... x 1)

x 1

1 2 .... (n 1) n

n (n 1) 2

57

lMhat;TI6 cUrKNnalImItxageRkam ³ x k> lim x x 4 3 1 x x x> lim xx (nx 1)xx 1n , n , p IN * dMeNaHRsay KNnalImItxageRkam ³ x k> lim x x 4 3 1 x x

4 x 1 3 2 n 1 x 1 p 1 p 4 x 1 3 2

( x 4 x ) 3( x 1) x ( x 1)( x 2 x 1) 3( x 1) lim 2 lim x 1 x ( x 1) ( x 1) x 1 ( x 1)( x 2 1) x3 x2 x 3 ( x 3 1) ( x 2 1) ( x 1) lim lim x 1 x 1 x2 1 ( x 1)( x 1) ( x 2 x 1) ( x 1) 1 3 2 1 6 lim 3 x 1 x 1 2 2 4 x 4x 3 lim 3 3 x 1 x x2 x 1 x n 1 (n 1) x n lim p1 , n , p IN * x 1 x xp x 1 ( x n 1 x ) n ( x 1) x ( x 1)( x n 1 ... x 1) n ( x 1) lim p lim x 1 x 1 x ( x 1) ( x 1) ( x 1)( x p 1)

dUcenH x>

.

x n .... x 2 x n ( x n 1) .... ( x 2 1) ( x 1) lim lim x 1 x 1 xp 1 xp 1 ( x 1)( x n 1 ... x 1) ... ( x 1)( x 1) ( x 1) lim x 1 ( x 1)( x p1 .... x 1) ( x n 1 ..... x 1) .... ( x 1) 1 n .... 2 1 n (n 1) lim x 1 x p1 .... x 1 p 2p x n 1 (n 1) x n n (n 1) lim p1 x 1 x xp x 1 2p

dUcenH

.

58

lMhat;TI7 cUrKNnalImItxageRkam ³ k> lim (1 ax) 1 x x> lim (1 ax) (1 bx) x K> lim (1 ax) (1x bx) 1 , n , p IN * , a, b IR dMeNaHRsay KNnalImItxageRkam ³ k> lim (1 ax) 1 x

n x 0 n p x 0 n p x 0 n x 0

(1 ax) 1 (1 ax ) lim

x0

n 1

.... (1 ax ) 1

lim a (1 ax ) n 1

x0

x .... (1 ax ) 1 a 1 1 ..... 1 n.a

dUcenH x>

dUcenH K>

(1 ax ) n 1 na lim x 0 x (1 ax ) n (1 bx ) p lim x 0 x n (1 ax) 1 (1 bx) p 1 lim x 0 x n (1 ax ) 1 (1 bx ) p 1 lim lim an bp x 0 x 0 x x (1 ax ) n (1 bx ) p an bp lim x 0 x (1 ax ) n (1 bx ) p 1 lim , n , p IN * , a , b IR x 0 x n (1 ax ) (1 bx ) p 1 (1 ax ) n 1 lim x 0 x p (1 ax ) n 1 n (1 bx ) 1 bp an lim(1 ax ) lim x 0 x 0 x x

.

.

59

lMhat;TI8 KNnalImItxageRkam ³ k> lim 2x x52 3 x> lim xx 14 3 K> lim x 3xx 1 3x 1 16 X> lim 2xx 3 xx 26 g> lim x 2x11 3x x 1

2 x2 2 x2 3 2 x 1 2 x 3 x 1

c> lim xx82 q> lim 1 xx 1 x C> lim x x 60 x 60 Q> lim x 1 x 1 x

x 8 2 x 0 3 2 3 2 2 x2 3 2 3 x 0

3

KNnalImItxageRkam ³ k> lim 2x x52 3

2 x2

lim

x2

2x 2 5 3 ( x 2)( 2 x 2 5 3 ) 2( x 2) 2x 2 5 3 8 2 3

lim

x 0

2( x 2)( x 2) ( x 2)( 2 x 2 5 3 )

lim

x2

4 3 3

dUcenH x> lim

x2

2x 2 5 3 4 3 lim x2 x2 3 x2 4

.

x3 1 3 ( x 2)( x 2)( x 3 1 3) ( x 2)( x 2)( x 3 1 3) lim lim x 2 x2 x3 8 ( x 2)( x 2 2 x 4) ( x 2)( x 3 1 3) 4.6 2 lim x2 x 2 2x 4 12

60

K>

x 2 3x 3x 1 lim x 1 x 1 2 x 3x 3x 1. x 1 lim x 1 x 1 x 2 3x 3 x 1 lim

x 1

( x 1)( x 1) x 1 . 2 x 1 x 3x 3x 1 ( x 1)( x 1) x 2 3x 3x 1 2.2 1 2 2

lim x

1

dUcenH X> lim

lim

x 3

x 2 3x 3x 1 lim 1 x 1 x 1

.

x 2 16 x 2 x 3 2x 3 x 6 x 2 16 x 2 4 x 4 2 x 3 x 6 lim . x 3 2x 3 x 6 x 2 16 x 2 4( x 3)( 2 x 3 x 6) ( x 3)( x 2 16 x 2) 4( 2 x 3 x 6) x 2 16 x 2 4.6 12 10 5

lim

x 3

dUcenH g>

x 1

x 2 16 x 2 12 lim x 3 5 2x 3 x 6 x 1 3x 1 lim x 1 2x 1 x x 2 2 x 1 3x 1 2 x 1 x . lim x 1 4 x 2 4 x 1 x x 1 3x 1 x ( x 1)(2 x 1 x ) ( x 1)(4 x 1)( x 1 3x 1) x (2x 1 x ) 2 1 (4 x 1)( x 1 3x 1) 3.4 6 lim

x 1

.

lim lim

x 1

dUcenH

x 1 3x 1 1 6 2x 1 x

.

61

c> lim xx82

x 8

3

( x 8)(3 x 2 23 x 4) 1 1 1 lim 3 2 x2 x 2 3 x 4 4 4 4 12

3

lim x 2

x8

dUcenH q> lim

x 0

lim

x 8

x2 x8

.

x2

1 x2 1 x 2 [3 (1 x 2 ) 2 3 1 x 2 1] lim x 0 1 x2 1

x 0 3

lim 3 (1 x 2 ) 2 3 1 x 2 1 1 1 1 3

dUcenH x C> lim x

x2

lim 3 x 0

3 2

x2 1 x2 1

3

.

x 2 60

3 x 2 60 4 2 2 2 2 x 6 x 2 60 x x 3 x 60 3 ( x 60) lim 6 . x2 x x 2 60 x 3 x 2 60 lim x2 x 4 x 2 3 x 2 60 3 ( x 2 60) 2 x 3 x 2 60 lim x2 x 3 x 2 60 16 16 16 48 3 88 16

dUcenH Q> lim

3 x 2 3 x 2 60 x 1 3 x 1 x 0 x x 1 1 (1 3 x 1) x 1 1 1 3 x 1 lim lim lim x 0 x0 x0 x x x x 11 1 x 1 lim lim x 0 x ( x 1 1) x 0 x (1 3 x 1 3 ( x 1) 2 )

.

lim x 0

1 1 1 1 5 lim 3 x 1 1 x 0 1 x 1 3 ( x 1) 2 2 3 6

62

lMhat;TI9 cUrKNnalImItxageRkam ³

6 x 4 12x 3 x 2 3 x 3 x 2 60 A lim 6 x2 x2 x 2 3 x 2 60

dMeNaHRsay KNnalImItxageRkam ³

6x 4 12x 3 x 2 3 x 3 x 2 60 A lim 6 x2 x2 x 2 3 x 2 60

tag U 6x

3

4

12 x 3 x 2 6 x 3 ( x 2) ( x 2) ( x 2)(6 x 3 1)

2

V x x 60

x 6 x 2 60 x 3 x 2 60

W x 3 x 2 60

Edl W x tag T x

2

6

x 2 60 ( x 6 64) ( x 2 4)

(x 2 4)(x 4 4x 2 16) (x 2 4)

(x 2)(x 2)(x 4 4x 2 15)

x 60

3 2

x 6 x 2 60

x 4 x 2 3 x 2 60 3 ( x 2 60) 2 W

3 2 2

x x

4

23

x 60 ( x 60)

2

.

3 U V 6 A lim x 2 x2 T

lim x 2

6

U V2 . 3 x2 T

63

3 (x 4 x 2 3 x 2 60 3 (x 2 60) 2 ) 3 W2 (x 2)(6x 1) . lim 6 x 2 x2 W3 (x 3 x 2 60) 2 (x 2)(6x 3 1)(x 4 x 2 3 x 2 60 3 (x 2 60) 2 ) 3 lim 6 x 2 (x 2)(x 3 x 2 60) 2 (x 2)(x 2)(x 4 4x 2 15) (6x 3 1)(x 4 x 2 3 x 2 60 3 (x 2 60) 2 ) 3 lim 6 x 2 (x 2) 2 (x 3 x 2 60) 2 (x 4 4x 2 15) 3 6 47.48 16.162.47 3

dUcenH

6x 4 12x 3 x 2 3 x 3 x 2 60 3 A lim 6 x 2 x2 x 2 3 x 2 60

64

lMhat;TI10 KNnalImIténGnuKmn_xageRkamenH

A lim x 1 x2 1 x3 1 x 4 1

x 1 x 1 x2 1

dMeNaHRsay KNnalImIt ³

A lim x 1 x2 1 x3 1 x 4 1

x 1 x 1 x2 1

x3 1 x4 1 x3 1 x4 1 x 1 x2 1

( x 1)( x 1) lim

x 1

x 1 x2 1 x 3 ( x 1) ( x 1)( x 1) x3 1 x4 1 lim x 1 x ( x 1) ( x 1) x 1 x2 1 x3 ( x 1)[ x 1 ] 3 4 x 1 x 1 lim x 1 x ( x 1)(1 ) 2 x 1 x 1 x3 1 x 1 2 4 2 1 ( 4 2 1)( 2 1) x3 1 x4 1 2 2 lim x 1 x 1 2 2( 2 1) 1 1 2 x 1 x2 1 8 4 2 2 1 7 3 2 2 2

( x 1)

65

lMhat;TI11 cUrKNnalImItxageRkamenH k> lim 1 xcos x2x sin x> lim tan xx sin x sin sin 2 x K> lim 2 sin x sin 3x 3 x 2x X> lim 1coscos 2xcos 44xx cos 1 g> lim 1 cos(x cos x) dMeNaHRsay KNnalImIt ³ k> lim 1 xcos x2x sin

3 x 0 x 0 3 x 0 x 0 x 0 4 3 x 0

c> lim 1 cos x cos 2x x cos q> lim 11 cos 2xx 1 C> lim 2 x. tanxcos x 1 cos x Q> lim cos sin cos 3x x cos j> lim x1(1 cos 3x ) x

x 0 2 x 0 x 0 2 x 0 3 x 0

dUcenH x> eday

(1 cos 2 x )(1 cos 2 x cos 2 2 x ) lim x0 x sin x 2 2 sin x (1 cos 2 x cos 2 2 x ) lim x0 x sin x sin x 2 lim . lim(1 cos 2 x cos 2 2 x ) 2.1.3 6 x0 x x 0 1 cos 3 2 x lim 6 x 0 x sin x tan x sin x lim x 0 x3 sin x tan x sin x cos x. tan x cos x

.

66

lim

dUcenH K>

lim

tan x cos x. tan x tan x (1 cos x ) lim lim x 0 x0 x 0 x3 x3 x sin 2 tan x 2 2.1. 1 1 2 lim . lim x 0 x x 0 x 2 4 2 tan x sin x 1 lim x 0 x3 2 2 sin x sin 2 x lim x 0 3 sin x sin 3x

2 tan x sin 2 x3

x 2

.

dUcenH 2x X> lim 1coscos 2xcos 44xx cos

x 0

2 sin x 2 sin x cos x x 0 3 sin x (3 sin x 4 sin 3 x ) 2 sin x (1 cos x ) lim x 0 4 sin 3 x x x 4 sin x sin 2 sin 2 2 lim 2 lim 3 x 0 x 0 sin 2 x 4 sin x x sin 2 2 ) 2 . 1 . x 12. 1 .12 1 lim ( x 0 x 4 sin 2 x 4 4 2 2 sin x sin 2x 1 lim x 0 3 sin x sin 3x 4

.

lim

(1 cos 4x ) (1 cos 2 x ) x 0 (1 cos 2 x ) cos 2 x (1 cos 4 x )

dUcenH

2 sin 2 2 x 2 sin 2 x sin 2 2 x sin 2 x lim lim 2 x 0 2 sin 2 x 2 cos 2 x sin 2 2 x x 0 sin x cos 2 x sin 2 2 x sin 2 2x sin 2 x 4 1 3 x2 x2 lim x 0 sin 2 x sin 2 2 x 1 4 5 cos 2x. x2 x2 cos 2 x cos 4 x 3 lim x 0 1 cos 2 x cos 4 x 5

67

1 g> lim 1 cos(x cos x)

x 0 4

x x 1 cos(2 sin 2 ) 2 sin 2 (sin 2 ) 2 lim 2 lim 4 4 x 0 x0 x x x x sin 2 (sin 2 ) sin 4 2 . 2 . 1 2. 1 1 2 lim x0 x x 16 16 8 (sin 2 ) 2 ( )4 2 2

1 dUcenH lim 1 cos(x cos x) 1 . 8 c> lim 1 cos x cos 2x x

x 0 4 x 0 2

dUcenH > q>

(1 cos x ) cos x (1 cos 2 x ) x 0 x2 1 cos x cos x (1 cos 2 x ) lim lim 2 x 0 x0 x2 x (1 cos 2 x ) x 2 sin 2 2 2 lim 2 cos x sin x lim x 0 x0 x2 x 2 (1 cos 2 x ) x sin 2 2 2 . 1 2 lim sin x . cos x 2 lim x 0 x0 x 2 4 x 2 1 cos 2 x ( ) 2 1 1 2. 2. 2 2 2 1 cos x cos 2 x lim 2 x 0 x2 1 cos x lim x 0 1 cos 2 x 1 cos x 1 cos 2 x lim . x 0 1 cos 2 x 1 cos x lim

.

68

dUcenH C>

x 2 .1 cos 2 x lim x 0 2 sin 2 x 1 cos x x sin 2 2 2 . x . 1 .1 cos 2 x 1 lim x 0 x 2 sin 2 x 4 1 cos x 4 ( ) 2 1 cos x 1 lim x 0 1 cos 2 x 4 2 1 cos x lim x 0 x. tan x 2 1 cos x lim x 0 x tan x ( 2 1 cos x ) 2 sin 2

.

x 1 cos x 2 lim lim x 0 x tan x ( 2 1 cos x ) x 0 x tan x ( 2 1 cos x ) 2 sin 2 x 1 2 2 . x .1. 2 lim x 0 x tan x 4 2 1 cos x 8 ( )2 2 2 1 cos x 2 lim x 0 x. tan x 8 1 sin 2 cos x lim x0 cos x cos 3x 1 sin 2 x cos 2 x cos x cos 3x . lim x 0 cos x cos 3x 1 sin 2 x cos x sin 2

dUcenH Q>

.

sin 2 x (1 cos 2 x ) cos x cos 3x . lim x 0 x 3x x 3x 1 sin 2 x cos x 2 sin sin 2 2 2 sin 2 x cos x cos 3x . lim x 0 2 sin 2 x.sin( x ) 1 sin 2 x cos x lim x 0 sin x 2 x 1 cos x cos 3x 1 . . . x sin 2 x 2 1 sin 2 x cos x 2

69

lMhat;TI12 cUrKNnalImItxageRkam ³

A lim 1 cos x cos 2 x cos 3x x 0 x2 1 cos x cos 2 x cos 3x....... cos nx B lim x 0 x2

dMeNaHRsay KNnalImItxageRkam ³

1 cos x cos 2 x cos 3x x2 (1 cosx) cosx(1 cos2x) cosx cos2x(1 cos3x) lim x0 x2 x 3x 2sin2 2 cosx sin2 x 2 cosx cos2x sin2 2 2 lim 2 x0 x 3x 2x sin2 sin 2 sin x 2 2 ( 1 1 9) 7 2 lim 2 2 cosx 2 cosx cos2x. x0 x x2 4 4 x 1 cos x cos 2 x cos 3x....... cos nx B lim x 0 x2 (1 cos x) cos x(1 cos 2x) ... cos x cos 2x...cos(n 1)x (1 cos nx) lim x 0 x2 x nx 2 sin 2 2 cos x sin 2 x ... 2 cos x cos 2x...cos(n 1)x.sin 2 2 2 lim 2 x 0 x nx 2x sin 2 sin 2 2 cos x. sin x .... cos x cos 2x...cos(n 1)x. x 2 lim 2 x 0 x2 x2 x 1 n2 12 2 2 ..... n 2 n(n 1)(2n 1) 70 2 ( 1 ...... ) 2 . 4 4 4 12 A lim x 0

lMhat;TI13 KNnalImItxageRkam ³

1 cos n x A n lim x 0 x2 n cos x cos 2 x cos 3 x ...... cos n x B n lim x 0 x2

dMeNaHRsay KNnalImIt

1 cos n x A n lim x 0 x2 (1 cos x )(1 cos x cos 2 x .... cos n 1 x ) lim x 0 x2 x 2 sin 2 (1 cos x cos 2 x .... cos n 1 x ) 2 lim x 0 x2 x sin 2 2 . 1 (1 cos x cos 2 x ..... cos n 1 x ) 2 lim x 0 x 4 ( )2 2 1 n 2. .(1 1 1 ... 1) 4 2

dUcenH

1 cos n x n A n lim x0 x2 2

.

n cos x cos 2 x cos 3 x ...... cos n x B n lim x 0 x2

71

(1 cos x ) (1 cos 2 x ) (1 cos 3 x ) ..... (1 cos n x ) lim x 0 x2 n 1 cos n x 1 cos n x n lim lim 2 2 x 0 x 0 n 1 n 1 x x n 1 2 3 .... n n ( n 1) ( ) n 1 2 2 4

n

dUcenH

n cos x cos 2 x cos 3 x ...... cos n x n (n 1) B n lim 2 x 0 x 4

lMhat;TI14 KNnalImItxageRkam ³ k> x> K>

x 2 lim x 1 (1 x ) 2 cos x lim 2 2 x 4x 2 sin x cos x lim x 4x 4

1 sin 2 1 sin x x cos 2 x 2 x 3 1 tan x lim x 1 1 x 2

c> lim(4 x ) tan 4x q> lim( x ) tan x 2

2 x 2 x

1 C> lim 1 tan x x 2 sin

x 4

X> lim g>

Q> lim j>

3x x 3 2 sin x 3 x3 8 lim x2 cos x

72

dMeNaHRsay KNnalImItxageRkam ³ k> tag z 1 x naM[ x 1 z kalNa x 1 enaH z 0

z z 1 sin( ) 1 cos 2 2 lim 2 lim z 0 z 0 z2 z2 z z 2 sin 2 sin 2 2 2 4 2 lim 4 . lim z 0 z 0 z z2 16 8 ( )2 4 x 1 sin 2 2 lim x 1 (1 x ) 2 8 cos x lim 2 2 x 4x 2 z x x z x 2 2 2 cos( z) sin z 2 lim lim 2 z0 z 0 2 4z 4z 2 2 2 4( z ) 2 sin z sin z 1 1 lim lim . z0 4z 4z 2 z0 z 4 4z 4 x 2 lim x 1 (1 x ) 2

1 sin

dUcenH x> tag

.

naM[

kalNa

enaH z 0

73

dUcenH K>

tag

dUcenH

cos x 1 2 2 x 4x 4 2 sin x cos x lim x 4x 4 z x x z x z0 4 4 4 sin( z) cos( z ) 4 4 lim z 0 4( z ) 4 2 2 2 2 ( cos z sin z ) ( cos z sin z ) 2 2 2 lim 2 z 0 4z 2 sin z 2 sin z 2 lim lim z 0 4z 4 z 0 z 4 sin x cos x 2 lim x 4x 4 4

lim

.

naM[

kalNa

enaH

X> lim tag

2 1 sin x x cos 2 x 2 z x x z x 2 2 2 2 1 sin( z ) 2 1 cos z 2 lim lim z0 z 0 sin 2 z 2 cos ( z) 2

naM[

kalNa

enaH z 0

74

lim z0

2 1 cos z 1 cos z lim 2 sin 2 z . ( 2 1 cos z ) z 0 sin z ( 2 1 cos z ) 2 sin 2

z 2 lim 2 z0 sin z . ( 2 1 cos z ) z 2 1 1 1 2 2 . z .1. 2. . 2 lim z 0 z 2 sin 2 z 4 2 1 cos z 4 2 2 8 ( ) 2 2 1 sin x 2 lim x cos 2 x 8 2 x 3 1 tan x lim x 1 1 x 2 sin 2

dUcenH

.

g> tag z 1 x naM[ x 1 z kalNa x 1 enaH z 0

(1 z) 3 1 tan( z ) lim z0 1 (1 z ) 2 1 3z 3z 2 z 3 1 tan z lim z0 1 1 2z z 2 3z 3z 2 z 3 tan z lim z0 2z z 2 tan z z(3 3z z 2 ) 3 z lim z0 z( 2 z) 2 x 3 1 tan x 3 lim x 1 1 x2 2 x lim( 4 x 2 ) tan x 2 4

dUcenH c> tag z 2 x naM[ x 2 z kalNa x 2 enaH z 0

75

lim 4 ( 2 z ) 2 tan z0

(2 z) 4

z 4 16 z ) lim ( 4 z ) 4 . lim ( 4 z z 2 ) tan( z0 z 0 z 2 4 tan 4

dUcenH q> lim( x ) tan x 2

x

lim( 4 x 2 ) tan x2

x 16 4

.

tag z x naM[ x z kalNa , x enaH z 0

lim z tan z 0

dUcenH C>

tag

z z z lim z tan( ) lim z cot z lim 1 z 0 z 0 z 0 2 2 2 tan z x lim( x ) tan 1 x 2 1 tan x lim x 1 2 sin x 4 z x x z ,x z0 4 4 4 1 tan z 1 1 tan( z) 1 tan z 4 lim lim z0 z0 2 2 1 2 sin( z ) 1 2 ( cos z sin z) 4 2 2 2 tan z lim z0 (1 cos z sin z)(1 tan z ) tan z 1 z 2 lim 2. 2 z 0 1 cos z sin z (0 1)(1 0) ( )(1 tan z ) z z

.

naM[

kalNa

enaH

76

dUcenH Q>

1 tan x 2 x 1 2 sin x 4 3x lim x 3 2 sin x 3

lim

.

tag z 3 x naM[ x 3 z kalNa , x 3 enaH z 0

3( z) 3z 3 lim lim z 0 z 0 3 1 3 2 sin( z) 3 2( cos z sin z) 3 2 2 3z 3 3 lim lim 3 z 0 3 3 cos z sin z z 0 3 1 cos z sin z 0 1 z z 3x lim 3 x 3 2 sin x 3

dUcenH

.

77

lMhat;TI15 KNnalImItxageRkam ³ k> lim(x

x 2 2

x 2) tan

3

x

c> q>

x lim x 1 1 x 2

2 x3 lim x 1 (1 x ) 2 1 sin

tan

x> lim K> X> g>

1 x x 1 cos x 1 (2 x ) 2 lim x2 1 sin x x cos x lim x x x 3 lim x 3 x cos x3

C> lim 2 2x

x2

sin

x

x Q> lim(1 x ) tan x 1 j> lim(2x x ) cot 2x

2 x 1 2 x2

dMeNaHRsay k> lim(x x 2) tan x

2 x 2

tag

1 1 x x z 1 1 lim( 2 2) tan z 1 z z z 2 z

naM[

kalNa , x 2 enaH

z

1 2

78

tag

dUcenH x>

tag

tag

1 1 1 z u ,z u0 z 2 2 2 1 1 1 lim[ 2] tan ( u ) u 0 1 1 2 ( u)2 u 2 2 1 1 1 u 2( u ) 2 3u 2u 2 2 2 lim tan( u ) lim cot(u ) u 0 u 0 1 1 2 ( u)2 ( u) 2 2 2 3 2u u 1 12 lim . . 2 u 0 (0.5 u ) tan u 12 lim( x 2 x 2) tan x 2 x 1 x3 lim x 1 cos x 1 1 1 1 z x 1 ,x 1 z x 1 z 2 1 1 ( 1) 3 z lim 1 z cos z 2 1 1 1 u z z u ,z u0 2 2 2 1 1) 3 1 ( 1 1 1 u ( u ) 3 (1 u ) 3 2 2 lim lim 2 u 0 u 0 1 cos( u ) ( u ) 3 sin u 2 2 u

naM[

kalNa

enaH

.

naM[

kalNa

enaH

naM[

kalNa

enaH

79

1 3 3 1 3 3 u u2 u3 u u2 u3 2 8 4 2 lim 8 4 u 0 1 ( u )3 sin u 2 3 3 u 2u 3 2u 2 u 1 12 2 lim lim 2 . . u 0 u 0 1 1 3 ( u ) sin u ( u ) 3 sin u 2 2

dUcenH K>

tag

tag

dUcenH

12 1 x3 lim x 1 cos x 1 2 (2 x ) lim x2 1 sin x 1 1 1 z x ,x 2 z x z 2 1 (2 ) 2 z lim 1 z 1 sin z 2 1 1 1 u z z u ,z u0 2 2 2 1 (2 )2 (1 2u 1) 2 0.5 u lim lim u 0 u 0 (0.5 u ) 2 (1 cos u ) 1 sin( u ) 2 u ( )2 2 4u 1 4 8 lim 2 lim . 2 . 2 2 u 0 u 0 u (0.5 u ) 2 2 2 u (0.5 u ) 2 sin sin 2 2 2 2 (2 x ) 8 lim x 2 2 1 sin x

.

naM[

kalNa

enaH

naM[

kalNa

enaH

.

80

cos

x

X> lim tag t xx x t t ebI x t 2

lim

t 2

x x x

cos t ( t ) cos t lim t t ( 2 t ) 2 t

tag u t t u ebI 2 2

x x 1 x 4

u0 2 u ( u ) cos( u ) sin u 1 2 2 lim 2 . lim u 0 u 0 2 u 4 ( 2 u )

cos lim x

t

dUcenH g> lim

tag

.

x 3 x 3 x cos x3 x z x3

naM[ x 3z z kalNa , x 3 enaH z 2

3z 3 3( 2z ) lim z lim z z ( z ) cos z cos z 2 2

tag u z z u ebI 2 2

lim u 0

z

u0 2

3( 2u ) 6u lim u 0 ( u ) cos( u ) ( u ) sin u 2 2 2 12 u 6 . lim u 0 sin u u 2

81

dUcenH c>

lim x3

tan

12 x3 x cos x3

.

tag

x lim x 1 1 x 2 1 1 z x ,x 1 x z tan z z 2 tan z lim lim 2 z 1 z 1 1 z 1 1 2 z

naM[

kalNa

enaH z 1

tag u 1 z naM[ z 1 u kalNa , z 1 enaH u 0

(1 u ) 2 tan( u ) lim u 0 (1 u ) 2 1 (1 u ) 2 ( tan u ) (1 u ) 2 tan u lim lim . . u 0 u 0 2u u 2 2u u 2 tan x lim x 1 1 x2 2 2 1 sin x3 lim x 1 (1 x ) 2 1 1 z x 3 x3 z 1 ,x 1 z 4 1 sin 2z z 2 (1 sin 2z) lim lim 1 1 1 z z (4z 1) 2 2 4 (1 4 3) z

dUcenH q>

.

tag

naM[

kalNa

enaH

82

tag

u

kalNa

1 z 4 1 ,z 4

naM[ z 1 u 4 enaH u 0

dUcenH C>

tag

(0.25 u ) 2 [1 sin( u )] (0.25 u ) 2 (1 cos u ) 2 lim lim 2 u 0 u 0 (1 4u 1) 16u 2 u (0.25 u ) 2 .2 sin 2 2 lim 2 u 0 16u u sin 2 2 1 1 2 2 1 2 2 . . . lim(0.25 u ) . u 8 u 0 4 8 16 4 512 ( )2 2 2 1 sin 2 x3 lim x 1 (1 x ) 2 512 2x lim x 2 2 sin x 2 2 z x ,x 2 z 1 x z 2 2 z 2 lim z 1 lim z 1 z 1 sin z z. sin z

.

naM[

kalNa

enaH

tag

u 1 z

naM[

z 1 u

kalNa , z 1 enaH

u0

u (1 u ) sin( u ) 2 1 u 1 . 2 lim . u 0 1 u sin u

2 lim u 0

83

dUcenH

lim x2

2x 2 2 sin x

8 x x 1 2 4 x

.

Q> j>

lim(1 x 2 ) tan x 1

lim(2 x x 2 ) cot x 2

84

lMhat;TI16 eK[GnuKmn_ f (x) ax bx2 4 x cUrkMnt;cMnYnBit a nig b edIm,I[ lim f (x ) 20 ? dMeNaHRsay kMnt;cMnYnBit a nig b ³ ax bx 4 eyIgman f (x) x 2

3 2 x 2 3 2

a ( x 3 8) b( x 2 4) (8a 4b 4) ( x 2) a ( x 2)( x 2 2 x 4) b( x 2)( x 2) (8a 4b 4) x2 8a 4b 4 a ( x 2 2 x 4) b ( x 2) x2

edIm,I[ lim f (x ) 20 luHRtaEt nig RKan;Et 8a 4b 4 0 nig lima (x 2x 4) b(x 2) 20 b¤ 12a 4b 20 8a b 4 eyIg)anRbB½næ 12a44b20 (1()2) dksmIkarBIrenHGgÁnigGgÁeK)an 4a 24 naM[ a 6 yktémø a 6 CYskñúgsmIkar ¬!¦ eK)an 48 4b 4 naM[ b 13 . dUcenH a 6 , b 13 .

x 2 2 x 2

85

lMhat;TI17 eK[GnuKmn_ f (x ) x axx 1bx 2 k> cUrKNna lim f (x) cMeBaH a 1 , b 2 . x> kMnt;témø a nig b edIm,I[ lim f (x ) 10 . dMeNaHRsay k> KNna lim f (x) ³ cMeBaH a 1 , b 2 eyIg)an ³

5 2 x1 x 1 x1

x 5 x 2 2x 2 lim f ( x ) lim x 1 x 1 x 1 x 2 ( x 1)( x 2 x 1) 2( x 1) lim x 1 x 1 limx 2 ( x 2 x 1) 2 3 2 1 x 1

dUcenH x> kMnt;témø a nig b eyIgman f (x ) x axx 1bx 2

5 2

x 5 x 2 2x 2 1 lim f ( x ) lim x 1 x 1 x 1

.

( x 5 1) a ( x 2 1) b( x 1) (a b 3) x 1 a b3 x 4 x 3 x 2 x 1 a ( x 1) b x 1

edIm,I[ lim f (x ) 10 luHRtaEt a b 3 0 b¤ a b 3 (1) nig limx x x x 1 a (x 1) b 10 b¤ 5 2a b 10

x 1 4 3 2

86

x 1

naM[ 2a b 5 (2) . dksmIkar ¬!¦ nig ¬@¦ eK)an a 8 naM[ a 8 nig b 3 a 11 . dUcenH a 8 ; b 11 . lMhat;TI18 cUrKNnalImItxageRkam ³ ( 2 x 1) k> lim (4x 1)(x 1) c> limx (x 1) x(x 2) ... (x 10) 10 2 2 x> lim(x x ) (x x ) q> lim (1 x)(1 2x1)(1 3x)...(1 10x) x x 1 K> lim x 42xx 38x 1 C> lim 3x x 1 3x 8 X> lim 27x 432x3 x 1 Q> lim x x x x x 1 g> lim 4x 7x 4x 5x j> lim[ x 6x x 6x ] dMeNaHRsay KNnalImItxageRkam ³ x 1 k> lim (4x(2 1)(x) 1)

3 4

10 10 10 10

x

5

7

x

10

10

2

2

x

x

10

5

5

x 3

3

3

3

x

5

5

3

3

3

3

x

5

5

x

3

2

2

3

2

3

3

2

x

x

3

4

x

5

7

87

x 12 (2 lim x x 5 (4

1 4 ) 3 x

dUcenH x> dUcenH K>

dUcenH

1 7 1 ) x (1 7 ) x5 x 1 (2 3 ) 4 24 x lim 4 x 1 1 4.1 (4 5 )(1 7 ) x x 3 (2 x 1) 4 lim 4 x ( 4 x 5 1)( x 7 1) 2 2 lim ( x ) 2 ( x ) 2 x x x 4 4 lim x 2 4 2 x 2 4 2 8 x x x 2 2 lim ( x ) 2 ( x ) 2 8 x x x 2x 3 lim 3 3 x x 4 x 3 8x 3 1 3 x (2 ) x lim x 4 1 x 3 1 2 x 3 8 3 x x 3 2 2 2 x lim x 4 3 1 1 2 3 3 1 8 3 x2 x 2x 3 2 lim 3 3 x x 4 x 3 8x 3 1 3

.

.

88

X>

dUcenH g> lim 4x 7x

2 x

27 x 3 4 33 8x 3 1 lim x x 5 32x 5 1 4 1 x 3 27 3 3x 3 8 3 x x lim x 1 x x 5 32 5 x 4 1 3 27 33 8 3 x3 x 3 6 3 lim x 1 2 1 1 5 32 5 x 3 27 x 3 4 33 8x 3 1 lim 3 5 5 x x 32x 1

3

.

4 x 2 5x

4 x 2 7 x 4 x 2 5x x lim 4 x 2 7 x 4 x 2 5x 12 x lim x 7 5 x 4 x 4 x x 12 12 x lim 3 7 5 22 4 4 x x

dUcenH c>

lim 4 x 2 7 x 4 x 2 5x 3 x

x 10 ( x 1)10 ( x 2)10 ... ( x 10)10 lim x x 10 1010

89

1 2 10 x 10 x 10 (1 )10 x 10 (1 )10 .... x 10 (1 )10 x x x lim 10 x 10 x 10 (1 10 ) x 1 2 10 1 (1 )10 (1 )10 ...... (1 )10 x x x lim 1 1 1 ... 1 11 10 x 10 1 10 x

dUcenH q>

dUcenH C>

x 10 ( x 1)10 ( x 2)10 ... ( x 10)10 11 lim 10 10 x x 10 (1 x )(1 2 x )(1 3x )...(1 10 x ) lim x 1 x 10 1 1 1 1 x ( 1) x ( 2) x ( 3).....x ( 10) x x x lim x x 1 x 10 ( 10 1) x 1 1 1 1 ( 1)( 2)( 3)......( 10) x x x 1.2.3....10 10! lim x x 1 1 10 x (1 x )(1 2 x )(1 3x )...(1 10 x ) lim 10! 10 x 1 x 3x 5 x 5 1 lim x 3x 5 x 5 1

1 1 x (3 5 1 5 ) 5 x lim x lim x x 1 1 5 1 3x x x (3 5 1 5 ) 5 x x 3x x 5 1 1 x5 3 1 2 lim x 3 1 1 3 5 1 5 x 3 5 1

90

dUcenH Q> lim

x

3x 5 x 5 1 lim 2 5 5 x 3x x 1 x x x x

x x x x x x x x x x x x x x 1 x 1 x x3

.

lim

x

x (1 lim

x

lim

x ) x

x

x (1

x x ) x x 1 1 x 1 1 x 1 1 x3 1 1 1 1 2

x 1 lim

x

lim

1 x 1 x

x

dUcenH j> lim[

x

1 lim x x x x x 2

3

.

x 3 6x 2 3 x 3 6x 2 ]

lim

x 3 6x 2 x 3 6x 2 ( x 3 6 x 2 ) 2 3 ( x 3 6 x 2 )( x 3 6 x 2 ) 3 ( x 3 6 x 2 ) 2

x 3

12 x 2 lim x 6 2 3 6 6 6 6 3 x 6 (1 ) x (1 )(1 ) 3 x 6 (1 ) 2 x x x x 12 x 2 lim x 6 6 6 6 x 2 [ 3 (1 ) 2 3 (1 )(1 ) 3 (1 ) 2 ] x x x x 12 12 lim 4 x 6 2 3 6 6 3 6 2 1 1 1 3 (1 ) (1 )(1 ) (1 ) x x x x

dUcenH

lim[3 x 3 6 x 2 3 x 3 6 x 2 ] 4

x

.

91

lMhat;TI19 KNnalImItxageRkam ³ k> lim x 2nx 1 x x> lim x 2x 1 x 4x 1 ........

2 x 2 2 x

x 2 2nx 1 nx

dMeNaHRsay KNnalImIt k> lim x 2nx 1 x

2 x

lim

x 2 2nx 1 x 2

x 2 2nx 1 x 2nx 1 lim x 2n 1 x 2 (1 )x x x2 1 1 x (2n ) 2n 2n x x lim lim n x x 1 1 2n 1 2n 1 1) 1 x( 1 1 x x2 x x2

x

dUcenH x> lim

x

x

lim

x 2 2nx 1 x n

x 2 2x 1 x 2 4x 1 ........ x 2 2nx 1 nx

lim ( x 2 2 x 1 x ) ( x 2 4 x 1 x ) .... ( x 2 2nx 1 x )

x

n lim ( x 2 2nx 1 x ) x n 1

n 2 x n

lim ( x 2nx 1 x ) ( n ) 1 2 3 .... n

n 1

n 1

n ( n 1) 2

92

lMhat;TI20 KNnalImItxageRkam ³ k> lim x 3nx 1 x x> lim x 3x 1 x 6x 1 ...... dMeNaHRsay k> lim x 3nx 1 x

3 3 2 x 3 3 2 3 3 2 x 3 3 2 x

3

x 3 3nx 2 1 nx

lim lim

x 3 3nx 2 1 x 3 ( x 3 3nx 2 1) 2 x 3 x 3 3nx 2 1 x 2 3nx 2 1

3

x 3

3n 1 2 3n 1 3 ) x 3 x 3 (1 3 ) x2 x x x x 1 x 2 (3 n 2 ) x lim x 3n 1 2 3 3n 1 3 ) 1 1 ] x 2 [ 3 (1 x x x x3 1 3n 2 3n x n lim x 111 3n 1 2 3 3n 1 3 (1 3 ) 1 1 x x x x3

x

x 6 (1

dUcenH lim x 3nx 1 x n x> lim x 3x 1 x 6x 1 ......

x

3 3 2 3 3 2 x

3

3

2

3

x 3 3nx 2 1 nx

lim ( 3 x 3 3x 2 1 x ) (3 x 3 6 x 2 1 x ) ... ( 3 x 3 3nx 2 1 x )

x

n 3 3 n 2 lim ( x 3nx 1 x ) lim(3 x 3 3nx 2 1 x ) x n 1 n 1 x n n ( n 1) ( n ) 1 2 3 ...... n 2 n 1

93

lMhat;TI21 KNnalImItxageRkam ³ k> lim 1 2 3n ...... n x> lim 1 2 3n ...... n n 4 8 K> lim 1 2 27 ........... ( 2 ) 3 3 1 1 1 1 ...... lim X> 1.5 5.9 9.13 (4n 3)(4n 1) 1 1 g> lim 1.4.7 4.71.10 7.10.13 ........ (3n 2)(3n1 1)(3n 4) dMeNaHRsay KNnalImIt k> lim 1 2 3n ...... n

2 2 2 2 n 3

3 3 3 3 n 3

n

n

n

n

2

2

2

2

n

3

dUcenH x> lim 1 2 3n ...... n

3 3 3 n 3

n ( n 1)(2n 1) n 6n 3 1 1 1 1 n 3 (1 )(2 ) (1 )(2 ) n n lim n n 1.2 1 lim n n 6n 3 6 6 3 12 2 2 32 ...... n 2 1 lim 3 n n 3 lim

.

3

n 4

n 2 (n 1) 2 n lim n 4n 3 4 (n 1) 2 n n 2 2n 1 n 2 2n 1 1 lim lim lim n n 4n 4 n 4n 4n 2

94

dUcenH K>

dUcenH X> 1 1 tag S 11.5 51.9 9.113 ..... (4n 3)(4n 1) (4k 3)(4k 1)

n n k 1

13 2 3 33 ...... n 3 n 1 lim n n3 4 2 2 2 8 lim 1 ........... ( ) n n 3 3 27 2 n 1 1 ( ) n 1 1 2 3 3 lim lim 0 n n 3 2 2 1 1 3 3 2 2 8 ........... ( ) n 3 lim 1 n 3 3 27 1 1 1 1 lim ...... n 1.5 5.9 9.13 (4n 3)(4n 1)

.

eRBaH

1 1 1 n 1 1 1 ( ) ( ) 4k 1 4 k 1 4k 3 4k 1 k 1 4 4 k 3 1 1 1 1 1 1 1 1 ) (1 ) ( ) ( ) ... ( 4 5 5 9 9 13 4n 3 4n 1

n n n

n

1 1 1 4 4n 1

dUcenH lim S lim 1 1 4n1 1 1 . 4 4 1 1 g> lim 1.4.7 4.71.10 7.10.13 ........ (3n 2)(3n1 1)(3n 4) tag S 1.41.7 4.71.10 ................. (3n 2)(3n1 1)(3n 4)

n

n

1 k 1 (3k 2)(3k 1)(3k 4) n 1 1 1 (3k 1)(3k 4) k 1 6 (3k 2)(3k 1) 1 1 1 1 1 1 1 ( )( ) ... (3n 2)(3n 1) (3n 1)(3n 4) 6 1.4 4.7 4.7 7.10 1 1 1 6 4 (3n 1)(3n 4)

95

n

dUcenH

1 1 1 1 lim S n lim [ ] n n 6 4 (3n 1)(3n 4) 24

.

lMhat;TI22 cUrKNnalImItxageRkam ³ 1 k> lim 2 1 2 3 2 2 3 ......... (n 1) n 1 n. n 1 1 1 x> lim 1n 31 1 5 3 ..... 2n 1 2n 1

n n

1 11 111 ........ 111......111

K> lim 10 X> lim (1 x)(1 x )(1 x )..........(1 x ) , | x | 1 1 2 1 3 1 n 1 lim n g> lim 2 1. 3 1....... n 1 c> k(k 1) dMeNaHRsay KNnalImItxageRkam ³ 1 1 1 ......... lim k> 2 2 3 2 2 3 (n 1) n n. n 1 eyIgtag

n n

n

2

4

2n

n

3

3

3

n

p

n

3

3

3

n

p 1

k 1

n

n 1 1 1 1 ..... Sn 2 2 3 2 2 3 (n 1) n n n 1 k 1 (k 1) k k k 1

96

1 k) k 1 k . k 1( k 1 k 1 k k. k 1 k 1 n 1 1 1 1 1 1 1 (1 )( ) ..... ( ) k 1 2 2 3 n n 1 k 1 k 1 1 n 1 1 lim S n lim 1 1 n n n 1

n

n

dUcenH x> lim 1n eyIgtag ³

n

.

1 1 1 ..... 3 1 5 3 2n 1 2 n 1

Sn

1 1 1 1 ( ) ..... n 3 1 5 3 2n 1 2n 1 1 n 1 n k 1 2k 1 2k 1 1 n 2k 1 2k 1 n 2k 1 2k 1 2k 1 2k 1 k 1 2 n k 1

1 ( 3 1) ( 5 3 ) ( 7 5 ) .......... ( 2n 1 2n 1) 2 n 2n 1 1 1 1 1 ) ( 2 2 n 2 n n

dUcenH K> lim

n

1 1 1 2 2 lim S n lim n 2 n n 2 n 1 11 111 ........ 111......111

n

.

10

n

97

lim

. dUcenH X> lim (1 x)(1 x )(1 x )..........(1 x ) , | x | 1

2 4 2n n

9 99 999 ......... 999....999 n 9.10 n (10 1) (10 2 1) (10 3 1) ........... (10 n 1) lim n 9.10 n (10 10 2 10 3 ......... 10 n n ) lim n 9.10 n 10 n 1 10. n 10 n 1 10 9n 10 1 lim lim n n 9.10 n 81.10 n 10 10 9n 10 lim n n 81 81.10 81 1 11 111 ........ 111......111 10 n lim n n 10 81

1 x 2 1 x 4 1 x 8 1 x2 . . ............. lim n 1 x 1 x 2 1 x 4 1 x2

n

n 1

1 x2 1 lim n 1 x 1 x

2 n

n 1

, ( lim x 2

n

n 1

0 | x | 1 )

2n

dUcenH lim (1 x)(1 x )(1 x )..........(1 x ) 1 1 x , | x | 1 . 2 1 3 1 n 1 lim . ....... g> 2 1 3 1 n 1 2 3 n k tag P 2 1. 3 1...... n 1 k 1 ((k 1)(k k 1)) 1 1 1 1 k 1)(k k 1

4

3

3

3

n

3

3

3

3

3

3

n

3

n

2

n

3

3

3

3

2

k2

k2

k 1 k k 2 k 1 . . k k 1 k 2 k 1 k2

n 2 k 1 n k n k k 1 2 k k 1 k k2 k 1 k2 k2 n

1 2 n 2 n 1 2(n 2 n 1) . . n n 1 3 3n ( n 1)

98

dUcenH c> lim n k(k1 1)

n p n p 1 k 1

2(n 2 n 1) 2 lim Pn lim n n 3n ( n 1) 3

.

n p 1 1 lim n ( ) n k 1 p 1 k 1 k n 1 1 1 1 1 1 1 ) lim n (1 ) ( ) ( ) ...... ( n 2 2 3 3 4 p p 1 p 1 n n p 1 lim n 1 p 1 nlim n p 1 n p 1 p 1 n n 1 2 3 1 lim n . . .... nlim n 2 3 4 n 1 n 1

dUcenH

n p 1 lim n k (k 1) 1 n p 1 k 1

.

lMhat;TI23 cUrKNnalImIt ³

lim 2x x 4 . tan( ) x 1 x 2 2x 3

dMeNaHRsay KNnalImIt ³

lim 2x x 4 ) . tan( x 1 x 2 2x 3

99

eKman 2x L lim eK)an 1 x

8 3 x 4 1 2x 8 1 (2x 3) (8 3 ) 2x 3 2 2x 3 2 2x 3 2 2(2 x 3)

x 4 2x 8 3 ) lim tan x 2x 3 x 1 x 2 2 2( 2 x 3) 2x 8 3 . cot lim x 1 x 2 2(2 x 3) 8 3 8 3 . tan( 2

BIeRBaH tan 2 2(2x 3) cot 2(2x 3) . tag y 2(82x33) kalNa x enaH y 0 eK)an

L lim 4 2 x 2 3x y 8 8 lim . 8 3 x 0 1 x 2 tan y 8 3 3 8 y

2x 2x 1 y 2 x 2(2 x 3) y . cot y lim . . . . lim 2 2 x 1 x x (1 x ) y tan y x 1 x 2 8 3 tan y y 0 y0 y0

dUcenH

lim

2x 8 x 4 . tan( ) x 1 x 2 2x 3 3 8

.

100

lMhat;TI24 cUrKNnalImItxageRkam ³ e e lim k> sin 2x x> lim e e , a, b IR * x K> lim 2e e 3e 1 5 ... X> lim e e x e n

x x x 0

ax bx x0

x 2x x 0 3x x 2x nx x 0

c> q> lim e x ex C> lim xe sinxx tan x x x Q> lim e e coscoscos 3x 2x

2 sin x tan 3 x x0 3

2 x 2 x0 3

e x cos 2 x lim x0 x2

2

3 sin 2 x

x 0

2x2

g> dMeNaHRsay KNnalImItxageRkam ³ k> lim esin 2ex

x x x 0 x

(e x 1)(e 2 x 1)...(e nx 1) lim x 0 xn

j>

e x 3 2 cos 4x lim x 0 x sin x

2

dUcenH x>

1 e x 2x e lim e 1 lim x 0 sin 2 x x 0 e x sin 2 x e 2 x 1 2x 1 lim . . x 1 x 0 2 x sin 2 x e ex ex lim 1 x 0 sin 2 x e ax e bx , a , b IR * lim x 0 x

.

101

dUcenH K>

(e ax 1) (e bx 1) lim x 0 x e ax 1 e bx 1 .a lim .b a b lim x 0 x 0 ax bx ax bx e e lim ab x 0 x x 2x 2e 3e 5 lim x 0 e3x 1 2(e x 1) 3(e 2 x 1) lim x 0 (e 3 x 1) ex 1 e2x 1 2. 6. x 2x 2 6 8 lim x 0 e3x 1 3 3 3. 3x x 2e 3e 2 x 5 8 lim 3x x 0 e 1 3 e x e 2 x ... e nx n lim x 0 x x (e 1) (e 2 x 1) ........ (e nx 1) lim x 0 x e2x 1 e nx 1 ex 1 lim 2. ..... n. x 0 2x nx x

dUcenH X>

.

dUcenH g>

n ( n 1) 2 x 2x nx e e ... e n n ( n 1) lim x 0 x 2 (e x 1)(e 2 x 1)...(e nx 1) lim x 0 xn ex 1 e2x 1 e nx 1 lim .2 ......n 1.2.3......n n! x 0 x 2x nx 1 2 .......... n

.

102

dUcenH c>

(e x 1)(e 2 x 1)...(e nx 1) n ( n 1) lim n x 0 x 2 2 e x cos 2 x lim x 0 x2

2

.

e x (1 2 sin 2 x ) lim x 0 x2 2 2 e x 1 2 sin 2 x e x 1 sin 2 x lim lim 2 lim 1 2 1 2 2 x 0 x 0 x 0 x x x

dUcenH q>

e x cos 2 x lim 1 2 x 0 x 2 sin x tan 3 x e e lim x 0 x3 x

(e 2 sin x 1) (e tan 3 x 1) lim x 0 x ( x 2 1)

2

.

e 2 sin x 1 sin x 2 e tan 3 x 1 tan 3x 3 lim lim 2 3 5 . . 2 . . 2 x 0 x 0 2 sin x x x 1 tan 3x 3x x 1

dUcenH C>

e 2 sin x e tan 3 x 5 lim 3 x 0 x x 2 xe 2 x sin x tan x x lim x 0 x3 2 x 2 x (e 1) cos x tan x tan x lim x 0 x3 2 x 2 e 1 tan x (cos x 1) lim lim x 0 x 0 x2 x3 x sin 2 2 x 2 e 1 tan x 2 2 2( 1 ) 2 5 2 lim 2 lim . 2 x 0 x 0 2x 2 x x 2 2

.

dUcenH

xe 2 x sin x tan x x 5 lim x 0 x3 2

2

.

103

Q> lim

x0

e 3 sin x cos x cos 3x e 2 x cos 2x

e 3 sin

2

2

2

x 3x (1 2 sin 2 )(1 2 sin 2 ) 2 2 lim 2x 2 x 0 e 1 2 sin x 3x x x 3x 2 sin 2 4 sin 2 sin 2 e 3 sin x 1 2 sin 2 2 2 2 2 lim 2x 2 x 0 (e 1 2 sin x ) 3x x x sin 2 sin 2 sin 2 3 sin x 2 1 sin x e 2 2 2 4 2 . sin 2 3x 3. . 2 2 3 sin 2 x x x2 x2 x2 2 lim 2x 2 x 0 e 1 sin x 2. 2 2. 2x 2 x 9 1 3 2. 2. 0 3 4 7 4 4 2 2 4 4

x

2 2 2 2 2

lMhat;TI25 cUrKNnalImIténGnuKmn_xageRkam ³ 3 c> lim 2 4 k> lim ln(2x 1) ln(x 2) 4 3 5 2x 1 9x 5 lim lim ln( ) ln( x> 6x 4 4x x q> 3 5 K> lim 6e 5 C> lim 2x 1 2 ln(2e 1) 2e 3 e X> lim 4e 9e Q> limln(12x 1) 3ln(2x 5) 2 3e g> lim (2x 3)(e 1) j> lim x 1 ln(2e 1)

x x

x

x

x

x

3

5

x

x

x

5

3

x

x 1

x 1

x

x

x

x

x

2 x

2x 2x

6

2

x

2x

x

2 x 1

x

x

x

104

dMeNaHRsay KNnalImIténGnuKmn_xageRkam ³ k> lim ln(2x 1) ln(x 2)

x

2 x 1 2x 1 lim ln ln 2 ln xlim x x 2 x 2

dUcenH lim ln(2x 1) ln(x 2) 2 9x x> lim ln( 62xx 1 ) ln( 4x 5 4 x

x

3

5

x

5

3

dUcenH K> lim 6e 2e

x

x 3 (2 2 x 1 9x 5 lim ln lim ln 5 . 3 x x 6 x 4 4x x x 5 (6 1 5 2 3 9 5 x . x ln 2 . 9 ln 3 lim ln x 4 1 4 6 4 6 5 4 2 x x 2x 3 1 9x 5 5 3 lim ln( 5 ) ln( 3 ln 4 x 4x x 6x 4

3 5

1 ) x 5 (9 3 x . 4 ) x 3 (4 5 x

5 ) x5 1 ) x2

.

1 0 x e x lim

x x

5 3

dUcenH X>

5 5 ) 6 x e x lim e 63 lim x x 3 3 2 e x (2 x ) 2 x e e 6e x 5 lim 3 x 2e x 3 4e 2 x 9e 2 x lim x 2e 2 x 3e 2 x e x (6

eRBaH

.

.

105

4 9e 2 x 2x 4 9e 4 x 4 e lim 2 lim x 2 x 2 3e 4 x 2 2x 3e 2x e

eRBaH

x

lim e 4 x 0

.

dUcenH g> tag

4e 2 x 9e 2 x lim 2 x 2e 2 x 3e 2 x 2 x 1 lim (2 x 3)(e 1) x 2 y x x 1

.

ebI

enaH y 0

lim (2 x 3)(e y 1)

x y0

e y 1 2 e y 1 lim ( 2 x 3) lim ( 2 x 3).y. . x y x0 x 1 y y y0 4x 6 ey 1 lim 4.1 4 . lim x x 1 y 0 y

dUcenH c>

2 x 1 lim (2 x 3)(e 1) 4 x x x 3 4 lim x x 2 4 x

x

. eRBaH

3 1 lim ( ) x 0 , lim ( ) x 0 x 4 x 2

3x 3 4 ( x 1) ( )x 1 4 lim lim 4 1 x x x 1 x x 2 ( ) 1 4 ( x 1) 2 4

dUcenH q>

3x 4 x lim 1 x 2 x 4 x 3x 5 x lim x 3 x 1 5 x 1

x

.

3 lim 0 x 5

x

dUcenH

3x 3 5 ( x 1) ( )x 1 1 1 5 lim lim . 5 x 1 x x 5 3 5 x 1 3 ( ) x 1 1 5 ( x 1 1) 5 5 3x 5 x 1 lim x 1 x 3 5 x 1 5

eRBaH

.

106

.

C> lim 2x 1 2 ln(2e

x

x 2 x 1

x

1)

lim ln e 2 x 1 ln(2e x 1) 2 2x e e .e lim ln x lim ln x 1 2 ( 2e 1) 2 x 2 x e (2 x ) e e e lim ln ln( ) x 4 ( 2 1x ) 2 e

e dUcenH lim 2x 1 2 ln(2e 1) ln( 4 ) . Q> limln(12x 1) 3 ln(2x 5)

x x 6 2 x

lim x ln(12 x 6 1) ln(2 x 2 5) 3 1 ) 6 12 x 6 1 x ] lim lim x [ln x ln 2 3 5 (2 x 5) x 6 (2 2 ) 3 x 1 12 6 x ] ln(12 ) ln 6 x [ln( lim 5 2 (2 2 ) 3 x x 6 (12

dUcenH

lim ln(12 x 6 1) 3 ln(2 x 2 5) ln 6 x

.

107

lMhat;TI26 eK[GnuKmn_ f kMnt;RKb;cMnYnBit x Edl f (x) x x 4 . k-cUrbgðajfasmIkar f (x) 0 manb¤sCacMnYnBitmYysßitenAcenøaH 1 nig 2 . x-cUrbgðafaRKb;cMnYnBit x eKman

5

f ( x ) 2 ( x 1)( x 4 x 3 x 2 x 2)

K-KNnalImIt A lim f (xx )12 . dMeNaHRsay k-karbgðaj eKman f (x) x x 4 eK)an f (1) 1 1 4 2 nig f (2) 2 2 4 30 eday f (1).f (2) 60 0 tamRTwsþIbTtémøkNþal enaHmancMnYn c mYyenAcenøaHcMnYn 1 nig 2 Edl f (c) 0 . dUcenH smIkar f (x) 0 manb¤sCacMnYnBitmYysßitenAcenøaH 1 nig 2 . x-bgðafa f (x ) 2 (x 1)(x x x x 2) eKman f (x ) 2 x x 2

x 1 3 5 5 5 4 3 2 5

( x 5 1) ( x 1)

( x 1)( x 4 x 3 x 2 x 1) ( x 1) ( x 1)( x 4 x 3 x 2 x 1) ( x 1)( x 4 x 3 x 2 x 2)

108

dUcenH f (x ) 2 (x 1)(x x x x 2) . f (x) 2 A lim K-KNnalImIt x 1 eday f (x) 2 (x 1)(x x x x 2) eyIg)an

x 1 3

4

3

2

4

3

2

(x 1)(x 4 x 3 x 2 x 2) A lim x 1 (x 1)(x 2 x 1)

x 4 x3 x 2 x 2 6 lim 2 x 1 x2 x 1 3 f (x) 2 2 A lim 3 x 1 x 1

dUcenH

.

lMhat;TI27 eK[GnuKmn_

sin x cos x 2 ( x) 2 f (x) 4 2 2

ebI ebI

x

4

x

4

cUrsikSaPaBCab;énGnuKmn_ f Rtg;cMnuc dMeNaHRsay sikSaPaBCab;énGnuKmn_ f Rtg;cMnuc x sin x cos x eKman lim f (x ) lim

x 4 x 4 0

x0

4

?

2

4

( x) 2 4

109

tag t x naM[ x t . kalNa x enaH t 0 4 4 4 eK)an

sin( t ) cos( t ) 2 4 4 lim f ( x ) lim 2 t 0 t x 4

sin cos t sin t cos cos cos t sin sin t 2 4 4 4 4 lim 2 t 0 t 2 2 2 2 cos t sin t cos t sin t 2 2 2 2 lim 2 2 t 0 t t 2 2 sin 2 2 cos t 2 2 (cos t 1) 2 lim lim 2 2 2 t 0 t 0 t t t t sin 2 2 2 2 f ( ) . lim 2 t 0 t 2 2 4 ( ) 2

eday

2 lim f ( x ) f ( ) 4 2 x 4

naM[ f (x) CaGnuKmn_Cab;Rtg; x

0

4

.

lMhat;TI28 sin( x ) f (x) eK[GnuKmn_ kMnt;RKb; x 1 . 1 x etIeKGacbnøayGnuKmn_ f [Cab;Rtg;cMnuc x 1 )anb¤eT ?ebIGac cUrkMnt;rkGnuKmn_bnøaytamPaBCab;énGnuKmn_ f (x) Rtg;cMnuc x 1 dMeNaHRsay kMnt;rkGnuKmn_bnøaytamPaBCab;

3

0

0

110

eKman lim f (x) lim sin(x) 1 x tag t 1 x naM[ x 1 t . kalNa x 1 enaH t 0 sin( t ) lim f ( x ) lim eK)an 1 (1 t )

x 1 x 1 3

x 1 t 0 3

lim

sin( t ) t 0 1 1 3t 3t 2 t 3 sin( t ) lim t 0 t (3 3 t t 2 ) sin( t ) lim . t 0 t 3 3t t 2 3

kMnt; enaHeKGacbnøayGnuKmn_ f (x) eday [Cab;Rtg;cMnuc x 1. ebIeyIgtag g(x) CaGnuKmn_bnøaytamPaBCab;énGnuKmn_ f (x) Rtg;cMnuc x 1 enaHeKGacsresr ³

x 1

lim f ( x )

3

0

0

dUcenH

sin( x ) f (x) 1 x3 g(x ) f (1) 3

ebI x 1 ebI x 1

111

lMhat;TI29 2x x 4 eK[GnuKmn_ f kMnt;elI IR eday f (x ) x 2x 4 . k> cUrkMnt;cMnYnBit a, b, c, d edIm,I[ f (x ) ax b x cx2xd 4 . x> cUrTajrksmIkarGasIumtUteRTtrbs;ExSekag (c) tagGnuKmn_ y f (x) . dMeNaHRsay k> kMnt;cMnYnBit a, b, c, d edIm,I[ f (x) ax b x cx2xd 4 eKman f (x) 2x 2x 4 x x4

3 2 2

2

3

2

( 2x 3 16) x 12 x 2 2x 4 2( x 3 8) x 12 x 2 2x 4 2( x 2)( x 2 2 x 4) x 12 x 2 2x 4 x 12 2x 4 2 x 2x 4

dUcenH a 2, b 4, c 1, d 12 . x> TajrksmIkarGasIumtUteRTt eKman f (x ) 2x 4 x x 12 4 eday lim x x 12 4 0 2x 2x dUcenHbnÞat; d : y 2x 4 CasmIkarGasIumtUteRTténRkab c tag f

2 x 2

112

lMhat;TI30 x eK[GnuKmn_ f (x) 2x x72 4 . k-cUrKNnalImIt lim f (x) rYcTajrksmIkarGasIumtUtQrrbs; Rkab (c) tag[GnuKmn_ f . x-kMnt;bIcMnYnBit A, B, C edIm,I[ f (x) Ax B x C 2 cMeBaHRKb; x 2 K-TajrksmIkarGasIumtUteRTténRkab (c) tag f . dMeNaHRsay k-KNnalImIt lim f (x) 2x 7 x 4 lim f ( x ) lim eyIg)an x2 2x 7 x 4 lim f ( x ) lim dUcenH lim f ( x ) . nig x2 eday lim f (x ) naM[eKTajfabnÞat;mansmIkar x 2 GasIumtUtQrénRkab . C f ( x ) Ax B x-kMnt;bIcMnYnBit A, B, C edIm,I[ x2 2x 7 x 4 f (x) eKman x2

2

x 2

x 2

2

x 2

x 2

2

x 2

x 2

x2

x 2

2

2 x 2 4 x 3x 6 2 2 x ( x 2) 3( x 2) 2 x2 x2 2 2x 3 x2

dUcenH

A 2, B 3, C 2

.

113

K-TajrksmIkarGasIumtUteRTténRkab (c) tag f 2 2 lim eKman f (x) 2x 3 x 2 eday x 2 0 dUcenHbnÞat;smIkar y 2x 3 CaGasIumtUteRTténRkab (c) tag f

x

lMhat;TI31 x px q f (x) eK[GnuKmn_ x 2x 2 k> cUrbgðajfaGnuKmn_ f kMnt;)anCanicÞcMeBaHRKb; x IR . x> kMnt;cMnYnBit p nig q ebIeKdwgfaExSekag (c) tagGnuKmn_ f manbnÞat; y x 2 CaGasIumtUteRTt ehIykat;tamcMnuc A(2,4) . dMeNaHRsay k> bgðajfaGnuKmn_ f kMnt;)anCanicÞ ³ x eKman f (x ) x px q 2x 2 eday x 2x 2 (x 2x 1) 1 (x 1) 1 0, x IR dUcenH GnuKmn_ f kMnt;)anCanicÞcMeBaHRKb; x IR . x> kMnt;cMnYnBit p nig q edIm,I[bnÞat; y x 2 CaGasIumtUteRTténRkab (c) tag f luHRtaEt ³ limf ( x ) ( x 2) 0 . eday f (x) (x 2) xx pxx 2q (x 1) (p x1)x 2 (q2 2) x 2

3 2 2

3 2 2 2 2

x

3

2

2

114

2

2

eK)an naM[ p 1 . x x GnuKmn_ Gacsresr f (x) x 2x q . 2 müa:geToteday ExSekag (c) tagGnuKmn_ f kat;tamcMnuc A(2,4) enaHkUGredaenéncMnuc A RtUvepÞógpÞat;smIkar (c) . 84q 4q f ( 2) 4 naM[ q 4 . eK)an 442 2 dUcenH p 1, q 4 .

3 2 2

(p 1) x 2 (q 2) limf ( x ) ( x 2) lim p 1 0 x x x 2 2x 2

115

lMhat;Gnuvtþn_

!-edayeRbIniymn½ycUrRsayfa ³ !> lim(2x 3) 7 ^> lim( x2 x ) 2 &> lim (2x 7) 1 @> lim(x x 2) 10 #> lim 3xx11 2 *> lim(x 4x 2x 3) 5 x 2x lim lim(2 x) 5 $> (> x 1 3 1 lim(sin x ) %> !0> lim (0.25) 4 2 @-eRbIniymn½ycUrRsayfa lim ln(1 x) ln 2 . #-KNnalImItxageRkamenH ³ 4 16 x 27 lim lim !> x 3 ^> 2 4 @> lim(2 x 3 ) ( x 3 ) &> lim 77 71 x x 2 *> lim 4 23. 1 2 #> lim (1 x) x (3x 1) $> lim x x x 1x 3 (> lim 22 16 4 %> lim x x x412 !0> lim 4x.2 42xx x4x $- KNnalImItxageRkam ³ 1 n x 1 lim( lim , n IN !> x 1 ^> 1 x 1 x )

3 x2 x2 3 2 x2

x 3 3 2

x 1

x4

3

x

x 3

x 1

2

x

x

6

x 1

x 1

3

x

x 3

x2

x

x

x

2

2

x 0

x 0

x

3

x

x

x 0

2

x0

x

2

3

2 x 1 2x

x 1

x 1

3

x

2

x2

2

x2

x

2

n

x 1

x 1

n

116

@> ( x 1) ( nx 1) lim #> x

n x 0 2

n 1 x 1 2

x x 2 x 3 ... x n n lim x 1 x 1

&> lim(1 mx 1 nx ) *> lim 1 (1 x)(1 x2 x).....(1 nx)

x 1 m n

x 0

( $> lim x ( xn1)1) x n (> lim x 1 n )....( ! %> lim nxx (x 1)xx 1 1 !0> lim ((xx11)(xx22)......(xx np))np! )( %-kMnt;témø m IR edIm,I[ ³ lim x x mx2 4 0 ? bx ^-kMnt; a nig b edIm,I[ lim ax x 3 6 5 &-kMnt; a nig b edIm,I[ lim x ax 2bx 4 8 x *-KNnalImItxageRkam ³ x x2 x 2 2 lim $> 2 x 2 !> lim x 2 @> lim 2 x x33 x 6 %> lim x x x2 60

x 1

x 2 x 2 ... nx n

n(n 1) 2

n 1

n

x 1

p 1

p

x 0

2

x2

3

x 3

3

2

x2

2

x2

x 2

x 3

x2

3

#> (- KNnalImItxageRkam ³ 2 2 x 2 lim !> 2 x 2 1 lim ( x x 1 1) @> x

x 1 x2 x0

lim

3x 1 2 x32

%> ^> lim

x0

lim

x x ... x 1 1 x 3 5x 7 2 x 1

x4 4

3x 2 18 3x 4

117

#>

x 6 x 2 x 1 lim x 3 x3

&>

x 3 x 2 60 lim x2 x2

$> lim 2

x2

x2 x x2

*>

x 3 x 2 x 2 60 3 x 2 60 lim x2 x2

!0-KNnalImIt

lim

x2

2 2 2 ... 2 x 2

x2

!!-KNnalImIt [email protected] ³ !> lim (1 x )((11 x)x )(1

x2

3 x 1 3

lim

x x x ... x 2 2

x2

4

x)

^> &>

x 1 3 x 1 lim x0 x

4

@>

x 1 3 3x 1 lim x 0 x2

2x 3 3 x 1 lim x2 x2

2 x 1 3

x3 1 lim x 1 2x 3 2x 1 4 x

x 3 x 2 60

2 3 2 x2

#> ( $> lim x x 31x) 2

lim

x 0

*> lim x x 60 x lim (> 1 (1 x)(1 2 x)...(1 nx)

x0 n

x2

n

%> x 1 nx 1 !#-KNnalImItxageRkam ³ !> lim x xx 1 x 3

3 x 1 3

,n2

!0>

1 lim ( n 1 x n 1 x ) x0 x

$> lim x

x 1

( x 1) 2

3

3x 2

@> #>

1 2 x .3 1 3 x 1 7x2 1 2 lim lim x0 x 1 x x 1 ( x 1) n (nx 1) lim , n IN * 3 2 x0 1 x 1

%>

118

!$-KNnalImItxageRkam ³ 2x 3 x 1 lim !> x2 x @> lim x 21x 1

3 x 2

3 x 1 3 3

^> lim &> lim

x 1

x x 1 3 x 1

x 1 x2 1 x3 1 x2 1 x3 1 x4 1

x 0 4

#> lim x 1

x 0

x2

4

4x 1

*> (>

lim

x 8 3

x 1 3 x x4 x 2

$> lim x

x 1

( x 1) 2 3x 2 x3 8

2

3

lim 3

x 1

x2 3

x x 2 x 4 x 2

x 3 x 2 60 x 3 4 x 1

%> lim

x 2 3

x 4x

!0> lim

!%-cUrKNnalImItxageRkam ³ !> %> x 2x 1 4x 4 x x 60 lim lim $> x x 60 @> x 1 x lim #> x 1 x 1 x 8 !^-cUrKNnalImItxageRkam ³ 1 cos 2 x sin x cos x lim lim !> 1 tan x ^> sin x sin x

3 3 2

6 2

2 x 1 3x 1 5 x 2 4 lim x 1 x 1

x 1

x2

4x 2 4 2x 2 lim x 1 x 1

2

3

2

3

2

x 0 3

2

3

2

3

2

3

x

4

x0

2

3

@>

3 4 sin 2 x lim 2 cos x 1 x

3

&>

2 x x 2 sin 2 x lim x 0 cos x x 1

119

#> $>

cos x sin 2 x lim 4 cos 2 x 3 x

6

*> (>

cos x x 2 1 lim x 0 x 2 sin 2 x

lim

x

tan 4 x cot 4 x 2 lim 1 2 sin 2 x x

4

sin x cos x 1 3 tan x

4

%> !0> lim x 1 cos x !&-cUrKNnalImItxageRkam ³ !> lim 24 416 2 ^> lim x x xx 1...

x 0 2

1 8 sin 3 x lim 2 x 4 cos x 3

6

x 4 sin 4 x

3

x

3

n

x n

x 2

x

x 1

@>

2 x 2x 2 lim 2 x 1 x 1 4 x 2 2x 1

2 x 1 3x 4 1 3

2x

x x 2 x 1

&> *>

x n x 2 lim x 1 x 1

m

#> lim

x 0

(1 m x )(1 n x ) 4 lim x 1 x 1

$> lim e e e lim %> e 1

x x 0 3x

ex x 1 2 xe x x 2 1

x

(> !0> lim

x 0

(1 x m ) n (1 x n ) m 2 m n lim x 1 x 1

m

1 xn m 1 xn xn

!*-cUrKNnalImItxageRkam ³ !> lim

x 1

x n 1 (n 1) x n ( x 1) 2

@> $>

1 lim x 1 ( x 1) 2

m

nx n 1 1 x n n 1

#>

(n 1) x n 1 x n 1 n lim 2 x 1 ( x 1)

x n x lim x 1 x 1

120

!(- cUrKNnalImItxageRkam ³ sin 2 x. sin 4 x. sin 8 x lim !> x sin x sin 3 x sin 5 x lim @> x #> lim sin x sin x 2 x sin 3x $> lim sin 3xxsin 2 x %> lim sin sin 4 x x @0- KNnalImItxageRkam ³ !> lim 1 xcos x cos 2 x cos 4 x lim @> x 1 cos 2 x lim #> sin 4 x $> lim 1 cosxx cos 3x 1 cos x cos 2 x lim %> x @!- KNnalImItxageRkam ³ !> lim 1 sin x 1 sin x x

x 0 3

x 0

3 3 3 x0 3

2

3

x 0

5

3

2

x 0

6

^> &> 2x *> lim tan sin.sin 3x x (> lim sin x sin 2 x sin 3x..... sin nx x sin(tan 2 x ) tan(sin 4 x ) lim !0> x

3 2 x 0 5

sin 2 (sin 2 x 3 ) lim x0 x6 sin[sin(si n x )] lim x0 x

x 0

n

x0

x 0

2

x 0

2

3

x0

2

x 0

2

x 0

2

^> lim 2 cos x xcos 2 x 3 tan x sin x lim &> x 2 sin x sin 2 x lim *> x 1 3 cos 2 x 2 cos 3 x lim (> x 1 cos x cos 2 x cos 3 x... cos nx lim !0> x

x 0 2

x0

3

x0

3

x 0

2

x0

2

x0

^> &> *>

lim

x 0

1 cos x 1 cos 2 x

@> #> lim

x 0

cos x cos 2 x lim x 0 x2 1 2 cos x 3 x2

1 cos n x lim x0 x sin x 1 3 cos x 2 lim x 0 1 cos 3 2 x

121

$> %> @@- KNnalImItxageRkam ³ !> 1 cos 2 x lim @> ( x)

x 2

2 3 cos 2 x lim x 0 x2 1 cos(1 cos x ) lim x 0 x4

(> cos x cos 2 x ... cos nx n lim !0> x

x0 2

1 3 cos 2x lim x0 x2

lim

sin x x 1 1 x 2

^> x lim1 x . tan &> 2

2 x 1

x 2 lim x 1 1 x 3

cos

1 sin

x

#> lim (sin x cos x )

4 4

x 2

2

*> (>

x 2 lim x 1 (1 x ) 2

1 sin

lim( 4 x 2 ) tan

x2

$> %>

lim ( 2 x) tan x

x

2

x

x 3 8 tan x lim x2 2 x

!0>

lim

1 x x 1 cos x 1

@#-KNnalImItxageRkam ³ 1 cos x lim !> (1 x)

x 1 2

@>

lim

1 x x 1 sin x

#> $> @$-KNnalImIténGnuKmn_xageRkam ³ 3 sin x sin 3x x sin x lim lim !> ^> 2 sin x sin 2x x

x 0 3

1 x 3 tan x lim x 1 1 x

x lim 2 2 2 x x cos

x 0

122

@> &> *> #> lim 1 cosxx cos 3x $> lim 1 sin 2x x (> cos tan x sin x lim %> 1 cos(1 cos x ) !0> @%-KNnalImIténGnuKmn_xageRkam ³

x 0 2

x 0

lim

1 cos 2x x 0 x sin x

x 0

x sin x x 0 tan x 1 cos(sin x ) lim x 0 x sin x 1 cos 2 x lim x 0 x tan x 1 2 3 cos x 1 lim 2 x 0 x cos x lim

!> @> lim

x 0

lim

x 0

2 1 cos x sin 2 x

x cos x (1 x ) cos x

sin 3x 3 sin x

tan x ^> lim x sin x tan x x sin x cos x cos 3x lim &> 1 cos 2x

x 0

x 0

#> *> cos x cos 5x 1 cos x cos 2 x 2 2 2 cos x 2 lim (> 1 cos 2x cos x $> lim x 2n 1 1 x 1 x 2x cos 2 x lim %> lim !0> sin x 1 cos 2x x @^-KNnalImIténGnuKmn_xageRkam ³ !> lim(1 x ) tan 2x ^> lim cos x xcos)a sin( a 1 2 @> lim tanxx &> lim tanxsincosxtx 1

x 0

lim

1 x x cos x x

2

lim

x 0

1 cos x 1 cos 5x

x 0

x 0

2

x 0

3

x 0

2

n

2

x 1

x a

3

x 1

x

4

#> $> lim(x

x 2

x 2 lim x 1 (1 x ) 2

1 sin

2

x 2) tan

x

*> (>

arctan x 4 lim x 1 1 x

lim

x 1 2

arcsin x arccos x 1 x 2

123

%> !0> @&-KNnalImIténGnuKmn_xageRkam ³

2

lim

cos x 2 2 x 4x

lim

1 cos x x ( x ) 2

!> lim(1 x ) tan 1 x

x 1

3

^> &> *> (>

x lim x 2 4 x 2

x2 1 lim x 1 cos x 1

cot

cos

@> lim

x

x x sin 4 4 x

#> $> %>

x 2 lim x 1 1 x 2

cos

lim x 8 sin x x 2 2x

3

lim(1 x ) tna

x 1

x 3x 1

x lim 2 x 2 x 4x 4

x 1

1 sin

(1 x ) 2 lim x 1 1 sin x 1

!0>

lim

arccos x x 1

@*-KNnalImIténGnuKmn_xageRkam ³ x sin x 1 lim !> lim x sin x ^> x cos x @> lim 1 2xx tan x 4 &>> lim(3x 1) sin xx21 2x 3

x

x

x

2

x

#> *> x $> lim(3x 1) tan x 1 (> lim 2xx 1 cos 2x 1 3 2 %> lim x x 1 tan 36xx 54 !0> lim cos x 1 cos x 1 4

3

x 2 lim (2 x 3) cos x 2x 1 x

x2 x 1 lim cot( ) x 2 x 1 2x 3

2

x

x

x

2

124

x

@(-KNnalImIténGnuKmn_xageRkam ³ ( x 1) ( x lim !> #> lim (2x 3) 35 2) x x x $> lim @> lim x2x 3 x

2

3

2

x

2

x

5

x

3

x

x x x

#0-cUrKNnalImIténGnuKmn_xageRkam ³ !> lim ( x 3x 7 x 9x 4 ) @> lim ( 4x 3x 1 4x 9x ) #> lim ( x x x x x ) $> lim ( x 4x x 4x ) %> lim ( 4x 7x 3 x 5x 1 9x 4x 5) ^> lim 2x 3 4x 6x 1 &> lim ( x 4x 1 x 2) *> lim ( 4x 8x 1 8x 12x x 5) (> lim ( x 4x x 3x x 2x 3x 2) #!-cUrKNnalImIténGnuKmn_dUcteTA ³ !> lim ( x 3x x 4x x 5x 9x 8x ) @> lim( 8x 4x 3x 5 8x 6x 2x 1) #> lim( x 4 x 6x 4 ) $> lim( x 3x x 6x x 9x 9x 4x 3 )

2 2 x 2 2 x x 2 2 x 2 2 2 x 2 x 3 3 2 x 2 3 3 2 x 4 4 3 3 3 2 2 x 2 2 2 2 x 3 3 2 3 3 2 x 3 3 2 x 3 3 2 3 3 2 3 3 2 2 x

125

%> lim ( &> lim

x

x

4

x 4 8x 3 1 3 x 3 9 x 2 4 x 1) x x x x

x x x x x

2 2 2 2

*> l lim( x x x 2x ................ x nx n x 8x 1) (> lim ( x 2x 1 x 4x 1 ............ x nx 1 n x 1) !0> lim (x 2)(x 4)(x 6) x . #@-cUrKNnalImItxageRkam ³ 1 2 3 n 1 lim ( ......... ) !> n n n n @> lim 1 3 5 7 ............... (2n 1) 2n2 1 n 1

x 3 2 3 2 3 2 3 3 3 x

3

x

n

2

2

2

2

n

#> $> lim 1 1 1 ............. 21 2 4 8 %> lim 1 2 3 n.............. n ^> lim 11.4 41.7 7.110 .......... (3n 21 3n 1) )(

n n

2 n 1 3n 1 lim n 2 n 3 n

3

3

3

3

n

4

n

&>

*> (> lim 1 23 25 ................... 2n2 1 2

n 2 3 n

1 1 1 1 ................ lim n 1.3 2.4 3.5 n ( n 2) 1 1 1 1 3 3 ............. 3 lim 3 n 2 1 3 1 4 1 n n

4 44 444 ............. 444........444

!0> lim

(n )

126

n

10 n

##-cUrKNnalImItxageRkamenH 1 !> lim (1 1 )(1 1 )(1 16 ).................(1 21 ) 2 4 @> lim cos x.cos x .cos 2x .....................cos 2x 2 #> lim 1 1 x 1 2x 1 4x .......... 1 2x ,| x | 1 1 2 3 n lim (1 )(1 )(1 ).................(1 ) $> n n n n 1 1 %> lim ( 2 1. 3 1. 4 1............. n 1) 2 3 1 4 n 1 #$-cUrKNnalImItxageRkamenH !> lim 1 2 3n .............. n n 1 1 1 1 1 ............ lim @> n 4 9 n

n 2n

n

2

n

n

n

2

4

2n

n

2

2

2

2

3

3

3

3

n

3

3

3

3

n

n 3

3

3

3

2

.......... ........ #> n 2 1 1 $> lim n n 1 1 n 2 ........... n 1kn 1 a 2a 3a ( na ) lim cos cos cos ............. cos %> n n n n n #%-KNnalImItxageRkam ³ 1 2 3 n lim ............. !> 2! 3! 4! (n 1)! 1.1!2.2!3.3!............... n.n! lim @> ( n 1)! ......... #> lim C 2C 4C .......... 2C 1 8

2

n

1 lim 2 n n 1

1

1

2 n n

n

n

n

0 n

1 n

2 n

n n

127

n

n

$>

%> x f (x) ,x0 #^- eK[GnuKmn_ 1 x x x k> cUrRsayfa 2 f (x) x 2 3 n 1 lim f ( ) f ( ) f ( ) ......... f ( ) . x> KNna n n n n

2

k2 lim 4 k k 2 1 n k 1 n n! lim n n

n

n

2

2

2

2

#&-k> cUrRsayfa x> tag P ln(1 n1 )(1 n4 )(1 n9 ).........(1 n ) . n cUrKNnalImIt lim P . #*-eK[sIVúténcMnYnBit (U ) nig (V ) kMnt;eday ³

2 3 n 3 3 3

x2 x ln(1 x ) x , x 0 2

n

n

n

n

U 0 1 , V0 4 1 3 U n 1 U n Vn 2 2 , n IN Vn 1 U n 2Vn

k>cUrbgðajfa W U V CasIVútFrNImaRt rYcKNna W CaGnuKmn_én n . x-cUrbgðajfa t 2U 3V CasIVútefrRtUvkMnt; . K-Tajrk U nig V CaGnuKmn_én n rYcbBa¢ak;lImItvakalNa n .

n n n n n n n n n

128

#(-eK[sIVút (U ) kMnt;elI IN eday ³ U 1 , U 4 nig n IN: U U .U k-eKtag n IN :V ln( U ) . U cUrbgðajfa (V ) CasIVútFrNImaRt rYcKNnatY V CaGnuKmn_én n nig lim V . x-cUrbgðajfa V ln U rYcTajrk U CaGnuKmn_én n U nig lim U . $0-eK[sIVút (U ) kMnt;elI IN eday ³ 11 1 U 1 , U 1 nig U U U . 10 10 cUrKNna U CaGnuKmn_én n nig lim U ? $!> eK[sIVút (U ) kMnt;elI IN eday U U U 1 , U 2 nig U k-eKtag V ln( U ) , n IN . U cUrbgðajfa (V ) CasIVútFrNImaRt rYcKNnatY V CaGnuKmn_én n nig lim V . x-eKtag S V V V ... V . bgðajfa U U .e rYcTajrk U CaGnuKmn_én n nigbBa¢ak;lImIt lim U .

n 0 1

n2

n 1

n

n 1 n

n

n

n

n

n

n 1

n

k

n

k 0

0

n

n

n

0

1

n 2

n 1

n

n

n

n

n

0

1

3 n2

4 n 1 n

n 1 n

n

n

n

n

n

n

0

1

2

n

Sn 1

n

0

n

129

n

n

[email protected][GnuKmn_ f

n

(x )

kMnt;eday ³

f1 (x) 6 x 3 , f 2 (x) 6 6 x 3 , f 3 (x) 6 6 6 x 3,.....

Edl n IN * . f (x ) f (x) L lim L lim cUrKNnalImIt nig . x 3 x 3 $#-eK[GnuKmn_ f (x) kMnt;eday ³

f n ( x ) 6 6 6 ....... 6 6 x 3

1 1 x 3

n

n

x 3

n

f1 ( x ) 2 x 3 3 , f 2 ( x ) 2 x 2 x 3 3 , f 3 ( x ) 2 x 2 x 2 x 3 3,.....

f n ( x ) 2 x 2 x 2 x ....... 2x 2 x 3 3

L1 lim f1 ( x ) x 3 x 3

L n lim f n (x ) x 3 x 3

Edl n IN *

cUrKNnalImIt nig . $$-eK[GnuKmn_ f (x) kMnt;cMeBaHRKb;cMnYnBit x eday ³ f (1 x ) f (4 2 x ) (1 x )(4 2x ) . cUrKNnalImIt lim f (x) . $%-eK[GnuKmn_ f (x) EdlkMnt;eday f (x 1) f (3x 1) 3xx 2 . 1 cUrKNnalImIt lim f ( x) nig lim f (x ) . $^-eK[GnuKmn_ f (x) 2 sin xx sin 2x . cUrkMnt;témø n IN * edIm,I[lImIt lim f (x ) CacMnYnBit . $&-eKmanGnuKmn_ f (x) ax bx c Edl a 0, a, b, c IR nig a b c 1 . KNnalImIt lim f [ff((xx))]xx .

x 2

2 2

x 2

x5

n

x 0

2

x 1

130

$*-eK[GnuKmn_ f (x) tan xx cot x Edl n IN * . cUrkMnt;témørbs; n edIm,I[lImIt lim f (x) kMnt; . x 4 x 2 lim sin( ) tan( $(-KNna x 1 2x 1 ) 4 a %0-eK[sIVút (n 1) 1 Edl n IN * k> KNna Sn a 1 a 2 a 3 .... a n . x> KNna Lim S .

n

x 0

x

n

2

n

n

131

emeronTI3

edrIevénGnuKmn_

1-cMnYnedrIevénGnuKmn_Rtg´cMnucmYy ½ snµtfaGnuKmn_ f ( x ) kMnt´elIcenøa¼ I ehIy x CacMnYnBitenAkñúg cenøa¼ I nig h CacMnYnBitminsUnü Edl x h Carbs´ I . cMnYnedrIevénGnuKmn_ f Rtg´cMnuc x ( ebIman ) CalImItrbs´GRta kMeNInénGnuKmn_ f rvag x nig x h kalNa h xiteTArksUnüEdleKkMnt´sresr ½

0

0

0

0

0

f ' ( x 0 ) lim

f (x 0 h) f (x 0 ) h 0 h

2-PaBmanedrIev nig PaBCab´ ½ snµtfaGnuKmn_ f (x) kMnt´elIcenøa¼ I ehIy x CacMnYnBitenAkñúgcenøa¼ I nig h CacMnYnBitminsUnü Edl x h Carbs´ I . -cMnYnedrIeveqVgRtg´cMnYn x énGnuKmn_ f (x) kMnt´tageday

0

0

0

132

f ' ( x 0 ) lim

h 0

f (x 0 h) f (x 0 ) h

0

f (x ) -cMnYnedrIevsþaMRtg´cMnYn x f (x h) f (x ) kMnt´tageday f ' (x ) lim . h -edrIevénGnuKmn_ f (x ) Rtg´ x ( ebIman ) kMnt´tageday ½ f (x h) f (x ) f (x h ) f (x ) f ' ( x ) lim ehIy lim h h f (x h) f (x ) f (x h) f (x ) lim lim mankalNa . h h 3-GnuKmn_edrIev ½ k¿niymn&y -ebI f CaGnuKmn_mYykMnt´elIcenøa¼ I nigmanedrIev Rtg´RKb´cMnuc enAkñúgcenøa¼ I ena¼eKfaGnuKmn_ f man edrIevelIcenøa¼ I . -GnuKmn_EdlRKb´ x I pßMáncMnYnedrIevén f Rtg´ x ehAfa GnuKmn_edrIevén f EdleKkMnt´sresr f : x f ' (x) .

0 0 0 h 0

. énGnuKmn_

0

0

0

0

0

0

h 0

h 0

0

0

0

0

h 0

h 0

133

smIkarbnÞat´b¨¼

y

( C ) : y = f (x)

(T)

M

1

x 0 1

cMnYnedrIevénGnuKmn_ f (x ) Rtg´cMnuc x KWCaemKuN Ráb´Tisén bnÞat´b¨¼nwgExßekag (c) : y f (x ) Rtg´cMnuc manGab´sIus x x ehIysmIkarbnÞat´b¨¼ena¼kMnt´eday½ (T ) : y y f ' ( x ) ( x x ) . x¿rUbmnþénGnuKmn_edrIevmYYycMnYn GnuKmn_ edrIev !> y k y' 0 y' n x @> y x

0

0

0 0 0

n n 1

134

#> y 1 x $> y x %> y e ^> y a &> y ln x *> y sin x (> y cos x !0> y tan x !!> y cot x [email protected]> y arcsin x !#> y arccos x !$> y arctan x K-rUbmnþRKwHénedrIevGnuKmn_ GnuKmn_ !> y u @> y u #> y u.v $> y u v

x x n

1 x2 1 y' 2 x y' y' e x y' a x ln a y' 1 x

y cos x y' sin x

y' 1 1 tan 2 x 2 cos x 1 y' 2 (1 cot 2 x ) sin x 1 y' 1 x2 1 y' 1 x2 1 y' 1 x2

edrIev

y' n.u '.u n 1 y' u' 2 u

135

y ' u ' v v' u

u ' v v' u y' v2

u' %> y ln u y' u ^> y sin u y' u '.cos u &> y cos u y' u ' sin u *> y e y' u '.e y' u ' (1 tan u ) (> y tan u u' !0> y arcsin u y' 1 u u' y' !!> y arccos u 1 u u' [email protected]> y arctan u y' 1 u u' !#> y u y' u v' ln u v u 4-edrIevbnþbnÞab´ ½ «bmafaeKman f (x ) CaGnuKmn_manedrIevTI n elIcenøa¼ eKkMnt´edrIevbnþbnÞab´énGnuKmn_en¼dUcteTA½ -GnuKmn_edrIevTImYykMnt´tageday f ' (x) -GnuKmn_edrIevTIBIrkMnt´tageday f ' ' (x) -GnuKmn_edrIevTIbIkMnt´tageday f ' ' ' (x) -GnuKmn_edrIevTIbYnkMnt´tageday f (x) -------------------------------GnuKmn_edrIevTI n kMnt´tageday f (x ) .

u u 2 2 2 2 V V

I

( 4)

136

(n )

5-vismPaBkMenInmankMnt´ ½ RTwsþIbT ½ eK[ f (x) CaGnuKmn_manedrIevelIcenøa¼ I ebImanBIrcMnYnBit m nig M Edl x I m f ' ( x ) M ena¼RKb´cMnYnBit a nig b CaFatuBIrkñúcenøa¼ I Edl a b eKán ½ m ( b a ) f ( b) f (a ) M ( b a ) . «bmafa a I , x I eKman ½ -ebI a x eKán m(x a) f (x) f (a) M(x a) . -ebI x a eKán m(a x) f (a) f (x) M(a x) .

137

lMhat;TI1 eK[GnuKmn_ f kMnt;elI IR eday f (x) x 3x 1 k-edayeRbIniymn½ycUrKNnacMnYnedrIev f ' (2) . x-cUrsresrsmIkarbnÞat; (T) Edlb:HnwgExSekag (c) Rtg;cMnuc M manGab;sIus x 2 .

3 0

0

dMeNaHRsay k-edayeRbIniymn½yKNnacMnYnedrIev f ' (2) tamniymn½yeyIgGacsresr ³ f ( 2 h ) f ( 2) eday f (x) x f ' ( 2) lim h

h 0

3

3x 1

3(2 h ) 1 2 3 3(2) 1 h 0 h 8 12h 6h 2 h 3 6 3h 1 8 6 1 lim h 0 h h 3 6h 2 9h lim limh 2 6h 9 9 h 0 h 0 h

(2 h ) lim

3

dUcenH f ' (2) 9 . x-sresrsmIkarbnÞat; (T) b:HnwgExSekag (c) Rtg;cMnuc M tamrUbmnþ (T) : y y y' (x x ) eday x 2 eK)an y f (2) 8 6 1 3 nig y' f ' (2) 9 eK)an (T) : y 3 9(x 2)

0 0 0 0 0 0 0

138

b¤ dUcenH

(T ) : y 3 9 x 18

(T ) : y 9 x 15

naM[ .

(T ) : y 9 x 15

.

lMhat;TI2 eK[GnuKmn_ f kMnt;elI IR eday f (x) 1 x sinx k-edayeRbIniymn½ycUrKNnacMnYnedrIev f ' (1) . x-cUrsresrsmIkarbnÞat; (T) Edlb:HnwgExSekag (c) Rtg;cMnuc M manGab;sIus x 1.

2 0

0

dMeNaHRsay k-edayeRbIniymn½yKNnacMnYnedrIev f ' (1) tamniymn½yeyIgGacsresr ³ f (1 h ) f (1) f ' (1) lim eday f (x) 1 x h

h0

lim

h0

sin x sin( h ) sin 1 (1 h ) 2 ) (1 12

2

h sin h 1 1 2h h 2 11 0 lim h0 h sin h 2h h 2 lim 2 h sin h 2 0 1 1 lim h0 h0 h h

139

dUcenH

f ' (1) 1

.

x-sresrsmIkarbnÞat; (T) ³ tamrUbmnþ (T) : y y y' (x x ) eday x 1 naM[ y f (1) 2 nig y' f ' (1) 1 dUcenH (T) : y 2 1.(x 1) b¤ (T) : y x 1 .

0 0 0

0 0 0

lMhat;TI3 eK[GnuKmn_ f kMnt;eday f (x) 2(xx21) cMeBaHRKb; x 2 . k-edayeRbIniymn½ycUrKNnacMnYnedrIev f ' (3) . x-cUrsresrsmIkarbnÞat; (T) Edlb:HnwgExSekag (c) Rtg;cMnuc M manGab;sIus x 3 .

0

0

dMeNaHRsay k-edayeRbIniymn½yKNnacMnYnedrIev f ' (3) tamniymn½yeyIgGacsresr ³

f ' (3) lim h 0 f (3 h ) f (3) h 2(3 h 1) 2(3 1) (3 h ) 2 3 2 lim h 0 h 4 2h 4 2 4 2h 4 4 h 1 h lim lim lim 2 h 0 h 0 h 0 h h (1 h ) 1 h f ' (3) 2

140

dUcenH

.

x-sresrsmIkarbnÞat; (T) ³ tamrUbmnþ (T) : y y y' (x x ) ( eday x 3 naM[ y f (3) 23321) 4 nig y' f ' (3) 2 eK)an T : y 4 2(x 3) dUcenH T : y 2x 10 .

0 0 0 0 0 0

lMhat;TI4

eK[GnuKmn_ f (x) kMnt; nig manedrIevRtg;cMnuc x c . cUrRsaybBa¢ak;fa lim f (c h) f (c h) 4f ' (c).f (c) h

2 2 h0

?

dMeNaHRsay

RsaybBa¢ak;fa tag

eday

f 2 (c h ) f 2 (c h ) lim 4f ' (c).f (c) h 0 h 2 2 f (c h ) f (c h ) L lim h 0 h f (c h ) f (c h )f (c h ) f (c h ) . lim h 0 h f (c h ) f (c h ) lim limf (c h ) f (c h ) h 0 h 0 h f (c h ) f (c) f (c h ) f (c) f (c h ) f (c h ) lim lim h 0 h 0 h h

lim h 0 f (c h ) f (c) f (c h ) f (c) lim h 0 h h f (c h ) f (c) f c ( h ) f (c) lim lim h 0 h 0 h (h ) f ' ( c ) f ' ( c ) 2f ' ( c ) 141

nig

limf (c h ) f (c h ) f (c) f (c) 2f (c)

naM[ L 2f ' (c) 2f (c) 4f ' (c).f (c) . dUcenH lim f (c h) f (c h) 4f ' (c).f (c) . h

2 2 h 0

lMhat;TI5 eK[ f CaGnuKmn_kMnt;elI IR eday f (x) 2xx 35xx34 k-cUrKNnaedrIev f ' (x) . x-kMnt;témø x edIm,I[ f ' (x) 0 rYcKNnatémøelxén f (x) cMeBaH témø x Edl)an . K-cUrrksmIkarbnÞat; (T) b:HnwgExSekag (c) tag f Rtg;cMnuc x 2 .

2 2

dMeNaHRsay k-KNnaedrIev f ' (x) eKman f (x) 2xx 35xx34 tamrUbmnþ ( u )' u' vv v' u v x eK)an f ' (x) (2x 5x 4)' (x 3xx 3)3x( 3) 3x 3)' (2x (

2 2 2

2

2

2

2

5 x 4)

2

2

(4x 5)(x2 3x 3) (2x 3)(2x2 5x 4) (x2 3x 3)2 4x3 12x 2 12x 5x2 15x 15 4x3 10x2 8x 6x2 15x 12 (x2 3x 3)2 x2 4x 3 2 (x 3x 3)2

dUcenH

x 2 4x 3 f ' (x) 2 ( x 3x 3) 2

142

.

x-kMnt;témø x edIm,I[ f ' (x) 0 eK)an f ' (x) (xx 34xx33) 0 naM[ x 4x 3 0 c eday a b c 0 eKTajb¤s x 1 , x a 3 . dUcenH x 1 , x 3 . KNnatémøelxén f (x) ³ eKman f (x) 2xx 35xx34 5 cMeBaH x 1 eK)an f (1) 21.1 351.134 2 3 34 1 . 1 cMeBaH x 3 eK)an f (3) 23.3 35.3.334 18 1534 7 99 3 dUcenH f (1) 1 , f (3) 7 . 3 K-rksmIkarbnÞat; (T) b:HnwgExSekag (c) . . cMeBaH x 2 naM[ y f (2) 222 35.2234 8410 34 2 6 kUGredaenéncMnucb:HKW M (2,2) . tamrUbmnþ (T) : y y y' (x x ) cMeBaH x 2 eK)an y' f ' (2) (22 34..2233) 1 eK)an (T) : y 2 1.(x 2) b¤ y x dUcenH smIkarbnÞat; (T) b:HnwgExSekag (c) KW (T) : y x .

2 2 2 2 1 2

1 2

2

2

2

2

2

2

2

2

0

0

0

0

2

0

2

2

143

lMhat;TI6 eK[ f CaGnuKmn_kMnt;elI IR eday f (x) x x px1 q eK]bmafamancMnYnBit a Edl f (a ) 0 . p cUrbgðajfa f ' (a ) 2aa 1 ?

2 2 2

dMeNaHRsay p bgðajfa f ' (a ) 2aa 1 eKman f (x) x x px1 q eK)an f ' (x) (x px q)' (x

2 2 2 2

cMeBaH eK)an tambMrab;eKman naM[ a ap q 0 2 yk ¬@¦ CYskñúg ¬!¦ eK)an³

2

1) ( x 2 1)' ( x 2 px q) ( x 2 1) 2 (2 x p)( x 2 1) 2 x ( x 2 px q) f ' (x) ( x 2 1) 2 ( 2a p)(a 2 1) 2a (a 2 ap q ) f ' (a ) (1) xa 2 2 (a 1) a 2 ap q f (a ) 0 2 a 1

2

dUcenH

(2a p)(a 2 1) 2a (0) f ' (a ) (a 2 1) 2 (2a p)(a 2 1) 2a p 2 (a 2 1) 2 a 1 2a p f ' (a ) 2 a 1

.

144

lMhat;TI7 eK[ f CaGnuKmn_kMnt;eday f (x) (x 1 x ) Edl x IR nig n IN * . k-cUrKNnaedrIev f ' (x) rYcbgðajfa 1 x .f ' (x) n.f (x) . x-cUrRsaybBa¢ak;TMnak;TMng (1 x ).f ' ' (x) x.f ' (x) n .f (x) .

2 n 2 2 2

dMeNaHRsay k-KNnaedrIev f ' (x) eKman f (x) (x 1 x ) eK)an f ' (x) n.(x 1 x )'.(x

2 n 2

1 x 2 ) n 1

(1 x 2 )' .( x 1 x 2 ) n 1 n.1 2 2 1 x 2x .( x 1 x 2 ) n 1 n.1 2 2 1 x n. 1 x2 x 1 x2 n 1 x2

2

.( x 1 x 2 ) n 1

(x 1 x 2 ) n

2 n

n dUcenH f ' (x) 1 x .(x 1 x ) . n eKman f ' (x) 1 x .(x 1 x ) eday f (x) (x n eK)an f ' (x) 1 x .f (x) naM[ 1 x .f ' (x) n.f (x) .

2 n 2 2

1 x 2 )n

145

2

dUcenH 1 x .f ' (x) n.f (x) . x-RsaybBa¢ak;TMnak;TMng ³

(1 x 2 ).f ' ' ( x ) x.f ' ( x ) n 2 .f ( x )

2

eKman eK)an

eKman nig yk ¬@¦ nig ¬#¦ CYskñúgTMnak;TMng 1 eK)an ³

1 n.f ( x ) x. f ' x n f ' ' ( x ) n. 2 1 x

f (x) 1 x2 f ' ( x ) 1 x 2 ( 1 x 2 )' f ( x ) f ' ' ( x ) n. ( 1 x 2 )2 2x f ' ( x ). 1 x 2 .f ( x ) 2 2 1 x f ' ' (x) n 1 x2 f (x) 1 x 2 .f ' ( x ) x. 1 x 2 1 f ' ' ( x ) n. 1 x2 1 f (x) 1 x 2 .f ' ( x ) n.f ( x ) 2 .f ' ( x ) n 1 x2 1 x 2 .f ' ( x ) n.f ( x )

naM[ f ' (x) n.

3

naM[ (1 x ).f ' ' (x) n .f (x) x.f ' (x) dUcenH (1 x ).f ' ' (x) x.f ' (x) n .f (x) .

2 2

2

2

146

lMhat;TI8 eK[ f CaGnuKmn_kMnt;elI IR eday f (x) 4x 16x 9 . cUrRsaybBa¢ak;fa ³ (16x 9).f ' ' (x) 16x.f ' (x) 4f (x) .

2 2

dMeNaHRsay RsaybBa¢ak;fa ³ (16x 9).f ' ' (x) 16x.f ' (x) 4f (x) eKman f (x) 4x 16x 9 eK)an f ' (x) (4x 16x 9 )'

2 2

2

2 4 x 16 x 2 9 32x 16x 4 4 2 16x 2 9 16x 2 9 2 4x 16x 2 9 2 4 x 16x 2 9

eKTaj eK)an

eday naM[ ykTMnak;TMng 2 nig 3 CYskñúg ¬!¦ eK)an ³ dUcenH (16x 9).f ' ' (x) 16x.f ' (x) 4f (x) .

2

2( 4x 16x 2 9 ) 2 2 16x 2 9 2 4x 16x 9 . 16x 9 2f ( x ) f ' (x) f ( x ) 4 x 16x 2 9 2 16x 9 32x f ' ( x ). 16x 2 9 .f ( x ) 2 2 16x 9 f ' ' (x) 2 2 16x 9 2f ( x ) 1 (16x 2 9).f ' ' ( x ) 2. 16x 2 9.f ' ( x ) 16x. 2 16x 9 2f ( x ) 2 f ' (x) 16 x 2 9.f ' ( x ) 2f ( x ) 3 16x 2 9

4( 16x 2 9 4x )

eRBaH

147

lMhat;TI9 1 sin eK[ f CaGnuKmn_kMnt;eday f (x) 2 sin x x x cos kMnt;RKb; x IR . k-KNnaedrIev f ' (x) . x-KNnatémø f ' 0 nig f ' . 2 dMeNaHRsay k-KNnaedrIev f ' (x) 1 sin eKman f (x) 2 sin x xcos x eK)an ³

f ' (x) (1 sin x )' (2 sin x cos x ) (2 sin x cos x )' (1 sin x ) (2 sin x cos x ) 2 cos x.(2 sin x cos x) (cosx sin x)(1 sin x) (2 sin x cos x) 2 2 cos x sin x cosx cos2 x cosx sin x cosx sin x sin2 x 2 sin x cosx 2 cosx sin x (cos2 x sin 2 x) cosx sin x 1 (2 sin x cosx) 2 2 sin x cosx 2 x x dUcenH f ' (x) (2cossinxsincosx1) . x-KNnatémø f ' 0 nig f ' 2 0 0 eK)an f ' 0 (2cossin0sincos01) (21 0011)

2 2

2

0

nig

sin 1 0 11 2 2 2 f ' 2 9 2 ( 2 sin cos ) 2 (2 1 0) 2 2

cos

.

148

lMhat;TI10 cos eKmanGnuKmn_ f x 1 sinx x k-cUrKNnaedrIev f ' (x) nig f ' ' (x) x-KNnatémø f ' (0) nig f ' ' (0) . dMeNaHRsay k-KNnaedrIev f ' (x) nig f ' ' (x) cos eKman f x 1 sinx x eK)an f ' (x) (cos x)' (1 sin1x)sin(1x sin x)' (cos x) ( )

2

. dUcenH müa:geTot dUcenH x-KNnatémø f ' (0) nig f ' ' (0) 1 1 eK)an f ' 0 1 sin 0 1 0 1 nig f ' ' (0) (1 cos 00) (1 10) 1 . sin dUcenH f ' 0 1 nig f ' ' 0 1 .

2 2

sin x (sin 2 x cos 2 x ) sin x sin 2 x cos 2 x (1 sin x ) 2 (1 sin x ) 2 sin x 1 1 (1 sin x ) 2 1 sin x 1 f ' (x) 1 sin x 1 (1 sin x )' cos x f ' ' x ( )' 1 sin x (1 sin x ) 2 (1 sin x ) 2 cos x f ' ' (x) (1 sin x ) 2

149

lMhat;TI11 cUrKNnaedrIevénGnuKmn_ ³ f ( x ) sin x. sin(n 1) x nig Edl n CacMnYnKt;FmµCati .

n 1

g( x ) cos n 1 x. cos(n 1) x

dMeNaHRsay KNnaedrIevénGnuKmn_ ³ eKman f (x) sin x.sin(n 1)x eK)an f ' (x) (sin x)'.sin(n 1)x (sin(n 1)x)'.sin

n 1 n 1

n 1

x

(n 1) cos x. sinn x. sin(n 1)x (n 1).cos(n 1)x. sinn1 x

(n 1).sinn x.sin(n 1)x. cos x sin x. cos(n 1)x (n 1).sinn x. sin(n 1)x x (n 1).sinn x. sin(n 2)x

dUcenH f ' (x) (n 1).sin x. sin(n 2)x . eKman g(x) cos x. cos(n 1)x eK)an g' (x) (cos x)'.cos(n 1)x (cos(n 1)x)'.cos

n 1 n 1

n

n 1

x

(n 1) sin x. cosn x. cos(n 1)x (n 1).sin(n 1)x. cosn 1 x

(n 1). cosn x.sin x. cos(n 1)x sin(n 1)x. cos x (n 1). cosn x. sinx (n 1)x (n 1). cosn x. sin(n 2) x g ' ( x ) ( n 1). cos n x. sin(n 2) x

dUcenH

.

150

lMhat;TI12 eK[GnuKmn_ f (x) x mx1 4 x Edl x CacMnYnBit nig m Ca)a:ra:Em:Rt . k-cUrkMnt;témø m edIm,I[GnuKmn_ f (x) mantémøbrmaRtg;cMnuc x 2 x-cUrkMnt;témø m edIm,I[GnuKmn_ f (x) mantémøbrmaEtmYyKt; .

2 2

dMeNaHRsay k-kMnt;témø m edIm,I[GnuKmn_ f (x) mantémøbrmaRtg;cMnuc x 2 luHRtaEt f ' (2) 0 eKman f (x) x mx1 4 x 1 eK)an f ' (x) (x mx 4)' (x (x) (1x 1)' (x mx 4) )

2 2 2 2 2 2 2 2

(2 x m)( x 2 1) 2 x ( x 2 mx 4) ( x 2 1) 2 2 x 3 2 x mx 2 m 2 x 3 2mx 2 8x ( x 2 1) 2 mx 2 6x m f ' (x) ( x 2 1) 2 4m 12 m 12 3m x2 f ' 2 0 2 (4 1) 25

cMeBaH naM[

eK)an m 4 .

151

x-kMnt;témø m edIm,I[GnuKmn_ f (x) mantémøbrmaEtmYyKt;luHRtaEtsmIkar f ' ( x ) 0 smmUl mx 6 x m 0 manb¤sEtmYyKt; eBalKWRtUv[ m 0 .

2

lMhat;TI13 eK[GnuKmn_ f (x) ax x bx4 3 cUrkMnt;témøéncMnYnBit a nig b edIm,I[GnuKmn_ f mancMnucbrma EtmYyKt; ehIyExSekagrbs;vamanbnÞat; y 2 CaGasIumtUtedk .

2 2

dMeNaHRsay kMnt;témøéncMnYnBit a nig b eKman f (x) ax x bx4 3 2 eK)an f ' (x) (2ax b)(x x4) 4x(ax

2 2 2 2 2

2

bx 3)

2ax 3 8ax bx 2 4b 2ax 3 2bx 2 6 x ( x 2 4) 2 bx 2 (8a 6) x 4b ( x 2 4) 2

edIm,I[GnuKmn_ f (x) mantémøbrmaEtmYyKt;luHRtaEtsmIkar f ' ( x ) 0 manb¤lEtmYyKt; .

152

eday naM[ bx 8a 6x 4b 0 manb¤leTalEtmYyKt;luHRtaEt b0 . müa:geTotExSekagrbs;vamanbnÞat; y 2 CaGasIumtUtedkluHRtaEt lim f ( x ) 2 . eK)an lim f x lim ax x bx4 3 lim ax a 2 . x dUcenH a 2 , b 0 .

2 x 2 2 x x 2 x 2

bx 2 8a 6 x 4b f ' x 0 x 2 42

lMhat;TI14 eK[GnuKmn_ f (x) x px q Edl x CacMnYnBitxusBIsUnü . x k-cUrKNnaedrIev f ' (x) nig f ' ' x . x-cUrkMnt;cMnYnBit p nig q edIm,I[GnuKmn_ f (x) mantémøGtibrmaesµI 1 cMeBaH x 2 .

2

dMeNaHRsay k-KNnaedrIev f ' (x) nig f ' ' x eKman f (x) x px q x p q RKb; x 0 . x x eK)an f ' (x) (x p q )' 1 xq nig f ' ' x (1 xq )' 2q x x

2 2 2 3

153

dUcenH f ' x 1 xq , f ' ' x 2q . x x-kMnt;cMnYnBit p nig q edIm,I[GnuKmn_ f (x) mantémøGtibrmaesµI 1cMeBaH x 2

2 3

luHRtaEt

eKTaj)an dUcenH

f ' 2 0 f 2 1 f ' ' 2 0 q f ' 2 1 0 4 4 2p q 1 f ( 2) 2 2q f ' ' 2 8

naM[

q 4 p 5 q 4 f ' ' 2 1 0 4 4

p 5 ,q 4

.

lMhat;TI15 b eK[GnuKmn_ f (x) ax 1 x Edl x CacMnYnBitxusBIsUnü . k-cUrKNnaedrIev f ' (x) nig f ' ' x . x-cUrkMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGb,brma esµI 5 cMeBaH x 1. dMeNaHRsay k-KNnaedrIev f ' (x) nig f ' ' x

154

b eKman f (x) ax 1 x b eK)an f ' (x) (ax 1 x )' a xb nig f ' ' x (a xb )' 2b x dUcenH f ' x a xb , f ' ' x 2b . x x-kMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGb,brmaesµI 5 cMeBaH x 1

2 2 3 2 3

luHRtaEt eK)an dUcenH

f ' 1 0 f 1 5 f ' ' 1 0 f ' 1 a b 0 f 1 a 1 b 5 f ' ' 1 2b a 2 ,b 2

naM[ .

a b 0 a b 4 f ' ' 1 2b

a 2 b 2 f ' ' 1 4 0

155

lMhat;TI16 eK[GnuKmn_ y f (x) x x ax1 b manRkabtMnag c kñúgtRmuyGrtUnremmYy. k-ExSekag c kat;GkS½Gab;sIus x'0x Rtg;cMnucmanGab;sIus x . a bgðajfabnÞat;b:H c Rtg; x manemKuNR)ab;Tis k 2 1 . x-cUrkMnt;témø a nig b edIm,I[ExSekag c kat;GkS½Gab;sIus )anBIrcMnuc M nig N EdlbnÞat;b:H c Rtg; M nig N EkgnwgKña .

2 2 2

dMeNaHRsay a bgðajfabnÞat;b:H c Rtg; x manemKuNR)ab;Tis k 2 1 eKman f x x x ax1 b eK)an f ' x (x ax b)' x (x1 (1x 1)' x ax b )

2 2 2 2 2 2 2 2 2

(2 x a )( x 2 1) 2x ( x 2 ax b) ( x 2 1) 2

ebI k CaemKuNR)ab;Tisén bnÞat;b:H c Rtg; x eK)an k f ' eK)an k 2 a (x (1) 21).( a b) 1 müa:geTotExSekag c kat;GkS½Gab;sIus x'0x Rtg;cMnucmanGab;sIus a b 0 naM[ a b 0 2 x eK)an f 1 a yksmIkar 2 CYskñúg 1 eK)an k (2( a)() 1) 2 1 1

2 2 2 2 2 2 2 2

156

2

2

2

a dUcenH k 2 1 . x-kMnt;témø a nig b smIkarGab;sIuscMnucRbsBV M nig N rvagExSekag c CamYy x'0x ³ x ax b f x 0 b¤ x ax b 0 E x 1 tag k nig k CaemKuNR)ab;TisénbnÞat;b:H cRtg; M nig N eK)an 2x a k f ' x nig k f ' x 2xx 1a x 1 ¬ tamsRmayxagelI ¦. edIm,I[bnÞat;b:HenHEkgKñaluHRtaEt

2 2 2 2 1 2 M N 1 M 2 M 2 N 2 N

2x a 2x N a k 1 .k 2 2M x 1 . x 2 1 1 M N

naM[ (2x

2x

M

a )(2x N a ) ( x 2 1)( x 2 1) M N

. a 2x N a (x 2 1)(x 2 1) 0 M N

M

4x M x N 2a(x M x N ) a 2 x 2 x 2 x 2 x 2 1 0 M N M N 4x M x N 2a(x M x N ) a 2 x 2 x 2 (x M x N )2 2x M x N 1 0 M N (x M x N )2 x 2 x2 2a(xM x N ) 2x M x N a 2 1 0 3 M N x M x N a xM xN E x M x N b

eday nig Cab¤ssmIkar enaHeKman TMnak;TMng 3Gacsresr ³

a 2 b 2 2a 2 2 b a 2 1 0 b 2 2b 1 b 1 0

2

naM[ b 1 müa:geTotedIm,I[ c kat;GkS½ x'0x )anBIrcMnuc M nig N luHRtaEt

157

smIkar E manb¤sBIrepSgKña eBalKWeKRtUv[ a 4b 0 eday b 1 eKTaj)an a 4 0 BitCanic©RKb;cMnYnBit a . dUcenH a IR , b 1 .

2

2

lMhat;TI17 eK[GnuKmn_ f (x) (ax b).e Edl a 0, a, b IR k-cUrKNnaedrIev f ' x nig f ' ' x x-kMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGtibrmaesµI 2e cMeBaH x 1 .

x

dMeNaHRsay k-KNnaedrIev f ' x nig f ' ' x eK)an f ' x (ax b)' e (e )' (ax b)

x x

ae x e x (ax b)

(ax a b).e x

nig

f ' ' ( x ) (ax a b)' e x (e x )' (ax a b)

a.e x e x (ax a b)

(ax 2a b).e x f ' x (ax a b)e x , f ' ' x (ax 2a b)e x

dUcenH

.

158

x-kMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGtibrmaesµI 2e cMeBaH x 1 luHRtaEt b¤

f ' (1) 0 f (1) 2e f ' ' (1) 0 2a b 0 a b 2 f ' ' 1 (3a b)e

eK)an naM[

f ' (1) (2a b)e 0 f (1) (a b)e 2e f ' ' (1) 0 a 2 b 4 f ' ' 1 2e 0

dUcenH

a 2 , b 4

.

lMhat;TI18 eK[GnuKmn_ f (x) axe b Edl a 0, a, b IR k-cUrKNnaedrIev f ' x nig f ' ' x x-kMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGb,brmaesµI e cMeBaH x 1.

x

dMeNaHRsay k-KNnaedrIev f ' x nig f ' ' x b eK)an f ' (x) (e )' (ax (ax)b()ax b)' e

x 2

x

e x (ax b) ae x (ax b a ).e x (ax b) 2 (ax b) 2

159

dUcenH

(ax b a ).e x f ' (x) (ax b) 2

.

[(ax b a )e x ]' (ax b) 2 [(ax b) 2 ]' (ax b a )e x f ' ' x (ax b) 4

[aex ex (ax b a)](ax b) 2 2a(ax b)(ax b a)ex (ax b) 4 (ax b) 2 .e x 2a(ax b a)ex (ax b) 2 2a(ax b a).ex (ax b)3 (ax b)3

. dUcenH x-kMnt;cMnYnBit a nig b edIm,I[GnuKmn_ f (x) mantémøGb,brmaesµI e cMeBaH x 1 luHRtaEt

f ' (1) 0 f (1) e f ' ' (1) 0

( ax b ) f ' ' x

2

2 a ( ax b a ) .e x ( ax b ) 3

bnÞab;BIedaHRsayeK)an

a 1 ,b 0

.

lMhat;TI19 eK[ f CaGnuKmn_kMnt;elI IR eday f (x) (x 2)(e 1) cUrsresrsmIkarbnÞat; (T) EdlRsbnwgbnÞat; d : y x 2 ehIyb:HnwgRkab (c) tMnagGnuKmn_ y f (x) .

x

dMeNaHRsay sresrsmIkarbnÞat; (T) ³

160

tag M (x , y ) CacMnucb:HrvagbnÞat; T eTAnwgExSekag (c) . tamrUbmnþ (T) : y y f ' x .(x x ) edayeKman T //d : y x 2 naM[eKTaj)an f ' (x ) 1 ¬emKuNR)ab;TisesµIKña¦. eKman f (x) (x 2)(e 1) eK)an f ' (x) (x 2)' (e 1) (e 1)' (x 2)

0 0 0 0 0 0 0 x x x

e x 1 e x ( x 2)

1 ( x 1).e x

x0 x0

eKTaj f ' (x ) 1 (x 1).e 1 b¤ (x 1).e 0 naM[ x 1 cMeBaH x 1 eK)an y f (1) (1 2).e e . smIkarbnÞat; T Gacsresr T : y e 1.(x 1) b¤ y x 1 e dUcenH (T) : y x 1 e .

0 0 0 0 0 0

161

lMhat;TI20 eK[ f CaGnuKmn_kMnt;eday f (x) sin x cos x k. sin x. cos x k-cUrKNnaedrIev f ' (x) . x-cUrkMnt;cMnYnBit k edIm,I[ f (x) CaGnuKmn_efrCanic©cMeBaHRKb; x IR .

6 6 2 2

dMeNaHRsay k-KNnaedrIev f ' (x)

f ' ( x ) 6.(sin x )' sin 5 x 6.(cos x )' cos 5 x k (sin 2 x )' cos 2 x k (cos 2 x )' sin 2 x 6 cos x sin 5 x 6 sin x cos 5 x 2k sin x cos 3 x 2k cos x sin 3 x

6 cos x sin x (sin 4 x cos 4 x ) 2k sin x cos x (cos 2 x sin 2 x ) 3 sin 2x (sin 2 x cos 2 x )(sin 2 x cos 2 x ) k sin 2 x. cos 2x 3 sin 2x ( cos 2 x ) k sin 2x cos 2 x 3 sin 2 x cos 2 x k. sin 2 x cos 2 x (3 k ). sin 2 x cos 2x 1 (k 3) sin 4x 2

dUcenH f ' (x) 1 (k 3).sin 4x . 2 x-kMnt;cMnYnBit k edIm,I[ f (x) CaGnuKmn_efrCanic©cMeBaHRKb; x IR luHRtaEt f ' ( x ) 0 RKb; x IR . eday f ' (x) 1 (k 3).sin 4x eKTaj k 3 0 naM[ k 3 .

162

lMhat;TI21 ln x f x ax b eK[GnuKmn_ Edl x 0 ehIy a nig b x CacMnYnBit. k-bgðajfacMeBaHRKb;cMnYnBit a nig b Edl a 0 ExSekag C tagGnuKmn_ f x manGasIumtUteRTtmYyEdleKnwgbBa¢ak;smIkar . x-kMnt;cMnYnBit a nig b edIm,I[ExSekag C tagGnuKmn_ f (x) b:HeTAnwgbnÞat; T : y x 4 Rtg;cMnuc A1,5 . dMeNaHRsay k-bgðajfaExSekag C tagGnuKmn_ f x manGasIumtUteRTtmYy ln x f x ax b eKman Edl x 0 x eday lim lnxx 0 naM[bnÞat; y ax b CaGasIumtUteRTt énExSekag C . dUcenH ExSekag C tagGnuKmn_ f x manGasIumtUteRTt y ax b . x-kMnt;cMnYnBit a nig b eKman f x ax b lnxx

x

163

eK)an f ' x (ax b)' (ln x)'.x x (x)'.ln x a 1 xln x . edIm,I[ExSekag C tagGnuKmn_ f x b:HeTAnwgbnÞat;

2 2

T : y x 4

Rtg;cMnuc A1,5 luHRtaEt ff ' xx ya naM[

A T A A

1 ln 1 a 12 1 a.1 b ln 1 5 1

b¤ a 1b15 b¤ ab 2 3 a dUcenH a 2 , b 3 .

lMhat;TI22 eK[GnuKmn_ f (x) x px2 q Edl x 2 . x k-cUrKNnaedrIev f ' (x) . x-cUrkMnt;cMnYnBit p nig q edIm,I[ f ' (x) x (x4x) 3 nig f (3) 3 . 2

2 2 2

dMeNaHRsay k-KNnaedrIev f ' (x) eKman f (x) x px2 q Edl x 2 x eK)an f ' (x) (x px q)' (x (2x)2()x 2)' (x

2

2

2

px q )

164

2

( 2x p)( x 2) ( x 2 px q ) ( x 2) 2 2 x 2 4 x px 2p x 2 px q x 2 4 x 2p q ( x 2) 2 ( x 2) 2

. dUcenH x-kMnt;cMnYnBit p nig q eday f ' (x) x (x4x) 3 nig f ' (x) x (4xx22p q 2 ) eKTaj x (4xx22p q x (x4x) 3 naM[ 2p q 3 (1) ) 2 nig f (3) 9 33p2 q 3 naM[ 3p q 6 (2) bUksmIkar ¬!¦ nig ¬@¦ eK)an p 3 bnÞab;mk q 6 3p 6 9 3 . dUcenH p 3 , q 3 .

2 2 2 2 2 2 2 2

x 2 4 x 2p q f ' (x) ( x 2) 2

lMhat;TI23 eK[GnuKmn_ f kMnt;elI IR eday f (x) sin x cUrbgðajfaedrIevTI n énGnuKmn_ f kMnt;eday f dMeNaHRsay bgðajfaedrIevTI n énGnuKmn_ f kMnt;eday f

(n )

( x ) sin( x

n ) 2

?

(n )

( x ) sin( x

n ) 2

165

eKman f (x) sin x eK)an f ' (x) cos x sin(x ) ¬ eRBaH 2

sin( ) sin 2

¦

f ' ' ( x ) ( x )' cos( x ) sin( x ) 2 2 3 f ' ' ' ( x ) ( x )' cos( x ) sin( x ) 2

]bmafavaBitdl;edrIevlMdab;TI n KW f (x) sin(x n2 ) Bit eyIgnwgRsayfavaBitdl;edrIevlMdab;TI (n 1) KW (n 1) f ( x ) sin x Bit 2 eyIgman f (x) (f (x))' eday f (x) sin(x n2 ) eK)an ³ n n n (n 1) f (x) (x )'cos(x ) sin(x ) sin x Bit 2 2 2 2 2 dUcenH f (x) sin(x n2 ) .

(n ) ( n 1) ( n 1) (n) (n )

( n 1)

(n)

166

lMhat;TI24 eK[GnuKmn_ f (x) 2x1 3 Edl x 3 2 cUrbgðajfaedrIevTI n énGnuKmn_ f kMnt;eday n!.2 f ( x ) (1) . (2 x 3)

n (n ) n n 1

dMeNaHRsay bgðajfaedrIevTI n énGnuKmn_ f kMnt;eday f (x) (1) eKman f (x) 2x1 3 tamrUbmnþ ( 1 )' vv' v 2 eK)an f ' (x) ((22xx 33))' (2x 2 3) (1) (21!. 3) x

(n ) 2 1 1 2 2

n

n!.2 n (2 x 3) n 1

2

]bmafavaBitdl;edrIevlMdab;TI KW eyIgnwgRsayfavaBitdl;edrIevlMdab;TI (n 1) KW (n 1)!.2 f ( x ) (1) Bit (2 x 3) eyIgman f (x) (f (x))' eday f (x) (1) eK)an f (x) (1) . n!.2(2[(2x3 3) ]' x )

n 1 ( n 1) n 1 n2 ( n 1) (n) (n ) n n 1 ( n 1) n 2 n 2 n 1

2[(2x 3) 2 ]' 8 2!.2 2 2 (1) f ' ' (x) 4 3 (2x 3) (2x 3) (2 x 3) 3 n!.2 n (n ) n f ( x ) (1) n (2 x 3) n 1

Bit

n

n!.2 n (2 x 3) n 1

dUcenH

n 1 n!.2 n.( n 1).2 n 1 ( n 1)!.2 (1) (1) ( 2x 3) n 2 (2 x 3) n 2 n!.2 n (n) n f ( x ) (1) (2x 3) n 1

Bit .

167

.

lMhat;TI25 eKmanGnuKmn_ f (x) (x 1).e Edl x CacMnYnBit . k> cUrKNnaedrIev f ' (x) , f ' ' (x) , f ' ' ' x nig f x . x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; f (x) (a x b ).e Edl a nig b CasIVútcMnYnBitepÞógpÞat;TMnak;TMng ³

2x (4) (n ) n n n n

2x

a n 1 2a n b n 1 a n 2b n , n IN *

K> cUrkMnt; a nig b CaGnuKmn_én n rYcTajrkkenSam f

n n

(n )

(x)

.

dMeNaHRsay k> KNnaedrIev f ' (x) , f ' ' (x) , f ' ' ' x nig eKman f (x) (x 1).e

2x

f ( 4 ) x

f ' ( x ) ( x 1)' e 2 x (e 2 x )' ( x 1)

e 2 x 2e 2 x ( x 1) e 2 x (1 2 x 2) (2x 3)e 2 x

dUcenH f ' (x) (2x 3)e eK)an f ' ' (x) (2x 3)' e (e

2x 2x

2x

2x

)' ( 2x 3)

2e 2 x 2e 2 x (2x 3) e 2 x (2 4 x 6) (4 x 8)e 2 x

dUcenH f ' ' (x) (4x 8)e eK)an f ' ' ' (x) (4x 8)' e (e

2x

2x

)' (4x 8)

168

4e 2 x 2e 2 x (4x 8) e 2 x (4 8x 16) (8x 20).e 2 x

dUcenH f ' ' ' (x) (8x 20)e eK)an f (x) (8x 20)' e (e

(4) 2x ( 4) 2x

2x

2x

)' (8x 20)

8e 2 x 2e 2 x (8x 20) e 2 x (8 16x 40) (16x 48)e 2 x

dUcenH f (x) (16x 48)e . x>bgðajfaedrIevTI n énGnuKmn_ f manTRmg; f (x) (a x b ).e tamsRmayxagelIeyIgman ³ Edl a 2 , b 3 f ' ( x ) (2x 3)e (a x b )e Edl a 4 , b 8 f ' ' ( x ) (4x 8)e (a x b )e f ' ' ' ( x ) (8x 20)e (a x b )e Edl a 8 , b 20 Edl a 16 , b 48 f ( x ) (16x 48)e (a x b )e >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ]bmafavaBitdl;edrIevlMdab;TI n KW ³ f (x) (a x b )e Bit eyIgnwgRsay[eXIjfavaBitdl;edrIevlMdab;TI (n 1) KW ³

(n ) n n 2x 2x 1 1 1 1 2x 2x 2 2 2 2 2x 2x 3 3 3 3 (4) 2x 2x 4 4 4 4 (n ) 2x n n

2x

f ( n 1) ( x ) (a n 1 x b n 1 )e 2 x

eyIgman eK)an

f ( n 1) ( x ) (f ( n ) ( x ))'

eday

f ( n ) ( x ) (a n x b n )e 2 x

f ( n 1) ( x ) (a n x b n )' e 2 x (e 2 x )' (a n x b n )

f ( n 1) ( x ) a n e 2 x 2e 2 x (a n x b n ) f ( n 1) ( x ) e 2 x (a n 2a n x 2b n ) f ( n 1) ( x ) ( 2a n x a n 2b n )e 2 x

169

eKTaj f (x) (a x b )e Bit Edl a 2a nig b a 2b . dUcenH edrIevTI n énGnuKmn_ f manTRmg; f (x) (a x b ).e Edl a nig b CasIVútcMnYnBitepÞógpÞat;TMnak;TMng a 2a nig b a 2b . K>kMnt; a nig b CaGnuKmn_én n tamsRmayxagelIeKman a 2a nig b a 2b cMeBaHRKb; n IN *. tamTMnak;TMng a 2a naM[ (a ) CasIVútFrNImaRtmanersug q 2 nigtY a 2 tamrUbmnþ a a .q 2.2 2 . müa:geToteKman b a 2b naM[eKTaj b 2b a 2 b¤ 21 b 21 b 1 ebIeKtag c 21 b eK)an c c 1 efr naM[ (c ) CasIVútnBVnþmanplsgrYm d 2 nigtY c b 3 tamrUbmnþ c c (n 1)d 3 2(n 1) 2n 1 eday c 21 b naM[ b 2 .c (2n 1).2 dUcenH a 2 , b (2n 1).2 .

n 1 n 1 n 1 n n 1 n n (n ) 2x n n n n n 1 n n 1 n n n n n 1 n n 1 n n n 1 n n 1 n 1 n 1 n n 1 n 1 n n n n 1 n n n n 1 n 1 n n n 1 n n 1 n n 1 1 n 1 n 1 n 1 n n 1 n n n n n 1 n n

( n 1)

2x

170

TajrkkenSam f (x) ³ eKman f (x) (a x b ).e eday a 2 nig b (2n 1)2 eK)an f (x) [2 x (2n 1)2 ].e 2 (x 2n2 1).e dUcenH f (x) 2 (x 2n2 1).e .

(n ) 2x n n n n n (n ) n n 1 2x n 2x (n) n 2x

(n )

n 1

lMhat;TI26 eKmanGnuKmn_ f (x) e . sin x k> cUrKNnaedrIev f ' (x) , f ' ' (x) , f ' ' ' x nig f x . x> cUrbgðajfaedrIevTI n énGnuKmn_ f manTRmg; ³

x (4)

f ( n ) ( x ) (a n sin x b n cos x ).e x

Edl a nig b CasIVútcMnYnBitepÞógpÞat;TMnak;TMng ³

n n

a n 1 a n b n b n 1 a n b n , n IN *

K> eKtag z a i.b . cUrbgðajfa z (1 i).z rYcsresr z CaTRmg;RtIekaNmaRt . X> cUrkMnt; a nig b CaGnuKmn_én n rYcTajrkkenSam f (x) .

n n n n 1 n n (n ) n n

171

dMeNaHRsay k> KNnaedrIev f ' (x) , f ' ' (x) , f ' ' ' x nig eKman f (x) e . sin x eK)an f ' (x) (e )' sin x (sin x)' e

x x x x

f ( 4 ) x

e x sin x cos x.e x e x (sin x cos x )

dUcenH f ' (x) e (sin x cos x) . eK)an f ' ' (x) (e )' (sin x cos x) (sin x cos x)' e

x

x

e x (sin x cos x ) (cos x sin x )e x

e x (sin x cos x cos x sin x ) 2e x cos x

x

dUcenH f ' ' (x) 2e cos x . eK)an f ' ' ' (x) 2(e )' cos x 2(cos x)' e

x x

x

2e x cos x 2 sin x.e x 2e x (cos x sin x )

dUcenH f ' ' ' (x) 2e (cos x sin x) . eK)an f (x) 2(e )' (cos x sin x) 2(cos x sin x)' e

(4) x

x

2e x (cos x sin x ) 2( sin x cos x )e x

2e x (cos x sin x sin x cos x ) 4e x sin x

(4) x

dUcenH f (x) 4e sin x . x> bgðajfaedrIevTI n énGnuKmn_ f manTRmg; tamsRmayxagelIeKman ³

x x

f (n) (x) (an sinx bn cosx).ex

172

f ' ( x ) e (sin x cos x ) (sin x cos x )e (a 1 sin x b1 cos x )e

x

Edl a 1 b 1

1 1

f ' ' ( x ) 2e x cos x (0. sin x 2 cos x )e x (a 2 sin x b 2 cos x )e x a 2 0 b 2 2

Edl Edl

f ' ' ' (x) 2e x (cos x sin x) (2 sin x 2 cos x)e x (a 3 sin x b3 cos x)e x a3 2 b3 2 f ( 4 ) ( x ) 4e x sin x (4 sin x 0. cos x )e x (a 4 sin x b 4 cos x )e x a 4 4 b 4 0

Edl >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ]bmafavaBitdl;edrIevlMdab;TI n KW f (x) (a sin x b cos x).e Bit eyIgnwgRsayfavaBitdl;edrIevlMdab;TI (n 1) KW

(n ) x n n

f ( n 1) ( x ) (a n 1 sin x b n 1 cos x )e x

eyIgman f

( n 1)

( x ) (f ( n ) ( x ))'

(a n sin x b n cos x )' e x (e x )' (a n sin x b n cos x )

f ( n 1) ( x ) (a n cos x b n sin x )e x e x (a n sin x b n cos x ) (a n cos x b n sin x a n sin x b n cos x )e x (a n b n ) sin x (a n b n ) cos x .e x f ( n 1) ( x ) (a n 1 sin x b n 1 cos x ).e x

Bit ¦

173

¬eRBaH a

a b

nig

b

a b

dUcenH edrIevTI n énGnuKmn_ f manTRmg; f (x) (a sin x b cos x).e Edl a nig b CasIVútcMnYnBitepÞógpÞat;TMnak;TMng ³

n n n n

(n )

x

a n 1 a n b n b n 1 a n b n , n IN *

K> bgðajfa z eKman z a

n

n 1

(1 i).z n

n

i.b n

sresr z CaTRmg;RtIekaNmaRt ³ naM[ z a i.b eday ³

n n 1 n 1 n 1

a n 1 a n b n b n 1 a n b n , n IN *

eK)an

z n 1 (a n b n ) i.(a n b n )

a n b n i.a n i.b n (1 i)a n (1 i)b n

(1 i)(a n

n 1 n

1 i b n ) (1 i)(a n i.b n ) (1 i)z n 1 i

dUcenH z (1 i)z . edayeKman z (1 i).z naM[ (z ) CasIVútFrNImaRténcMnYnkMupøic Edlmanersug ³ 2 2 q 1 i 2( i. ) 2 (cos i. sin ) nigtY z a ib 2 2 4 4 Et a 1, b 1naM[ z 1 i q 2 (cos i. sin ) . 4 4 tamrUbmnþtYTI n énsIVútFrNImaRteKGacsresr ³

n 1 n n 1 1 1 1 1 1

n 1

z n z1 q 2 (cos i.sin ). 2 (cos i.sin ) 4 4 4 4 ( 2 ) n (cos i.sin ) n 4 4

n 1

174

dUcenH

z n 2 n (cos

n

i. sin

n

)

¬ tamrUbmnþdWmr½ ¦ .

X> kMnt; a nig b CaGnuKmn_én n eKman z a i.b eday z 2 (cos n4 i. sin n4 )

n n n n n n n

eKTaj

a n i.b n 2 n (cos

n n i.sin ) 4 4

n 4

naM[ .

n a n 2 n . cos 4 b 2 n .sin n n 4

dUcenH a 2 . cos n4 TajrkkenSam f (x) ³

n n (n )

, b n 2 n . sin

f ( n ) ( x ) (a n sin x b n cos x )e x

a n 2 n . cos

eday

n n , b n 2 n . sin 4 4

eK)an dUcenH

f ( n ) ( x ) ( 2 n . cos

2 n (cos

n n x .sin x 2 n cos x.sin ).e 4 4

n ) 4

n n n sin x sin cos x ).e x 2 n . sin( x ).e x 4 4 4

f ( n ) ( x ) 2 n .e x sin( x

.

175

lMhat;TI27 eKmanGnuKmn_ f (x) 3x 1 kMnt;elI 1 , 3 3 k> cMeBaHRKb; x 1,5 cUrbgðajfa 8 f ' (x) 3 . 4 x> edayeRbIvismPaBkMeNInmankMnt;eTAnwgGnuKmn_ f cMeBaHRKb; 3 13 3 5 . x 1,5 cUrbgðajfa x 3x 1 x 8 8 4 4 dMeNaHRsay 3 k> cMeBaHRKb; x 1,5 bgðajfa 8 f ' (x) 3 4 eKman f (x) 3x 1 naM[ f ' (x) 2 33x 1 cMeBaHRKb; x 1,5 eKman 1 x 5 b¤ 4 3x 1 16

1 1 1 4 3x 1 2 3 3 3 8 2 3x 1 4 3 dUcenH 8 f ' (x) 3 cMeBaHRKb; x 1,5 . 4 3 x> bgðajfa 8 x 13 3x 1 3 x 5 8 4 4 3 cMeBaHRKb; x 1,5 eKman 8 f ' (x) 3 4 3 cMeBaH x 1 eKman 8 (x 1) f (x) f (1) 3 (x 1) Et f (x) 4 3 3 eK)an 8 x 8 3x 1 2 3 x 3 4 4 3 dUcenH 8 x 13 3x 1 3 x 5 . 8 4 4

3x 1

176

lMhatTI28 eK[GnuKmn_ y f (x) x 23xx 3 c k> cUrkMnt;bIcMnYnBit a, b nig b edIm,I[ f (x) ax b 2 x cMeBaHRKb; x 2 . x> cUrkMnt;smIkarGasIumtUtQr nigGasIumtUteRTténRkab c tMnagGnuKmn_ f . K> KNnaedrIev f ' (x) rYcKUstaragGefrPaBénGnuKmn_ f (x) . X> RsayfacMnuc I(2 , 1) Cap©itbMElgqøúHénRkab c . g> cUrsg;Rkab (c) tagGnuKmn_ f (x) kñúgtMruyGrtUnrm:al; (0, i , j) mYy

2

dMeNaHRsay c k>kMnt;bIcMnYnBit a, b nig b edIm,I[ f (x) ax b 2 x cMeBaHRKb; x 2 eKman y f (x) x 23xx 3

2

eK)an dUcenH

( x 2 2 x ) ( x 2) 1 2x x ( x 2) ( x 2) 1 1 x 1 2x 2x 1 c y f (x) x 1 f ( x ) ax b 2x 2x

eday

a 1, b 1, c 1

.

177

x> kMnt;smIkarGasIumtUtQr nigGasIumtUteRTt ³ eKman f (x) x 23xx 3 CaGnuKmn_kMnt;elI D IR {2} . eday lim f (x) lim x 23xx 3 nig lim f (x) lim x 23xx 3 naM[bnÞat; x 2 smIkarGasIumtUtQrénRkab (c) . 1 müa:geToteKman f (x) x 1 2 x 1 eday lim 2 x 0 naM[bnÞat; y x 1 CaGasIumtUteRTténRkab (c) . K> KNnaedrIev f ' (x) 3 eKman f (x) x x x2 3 CaGnuKmn_kMnt;elI D IR {2} ) eK)an f ' (x) (2x 3)(x x22)(x 3x 3) (

2

f

2

x 2

x 2

2

x 2

x 2

x

2

f

2

2

2 x 2 4 x 3x 6 x 2 3x 3 ( x 2) 2 x 2 4x 3 ( x 2) 2

. dUcenH KUstaragGefrPaBénGnuKmn_ f (x) ³ ebI f ' (x) x (x42x) 3 0 eK)an x 4x 3 0

2

2

x 2 4x 3 f ' (x ) ( x 2) 2

2

178

c naM[ x 1 , x a 3 cMeBaH x 1 eK)an f (1) 1 231 3 1 cMeBaH x 3 eK)an f (3) 9 9 3 3 23 1 eday f (x) x 1 2 x 1 eK)an lim f (x) lim (x 1 2 x ) eKGacKUstaragGefrPaBdUcxageRkam ³

1 2

2

1

2

x

x

x

f ' (x) f (x)

1

2

3

1

3

X>RsayfacMnuc I(2 , 1) Cap©itbMElgqøúHénRkab c 1 eKman f (x) x 1 2 x tamrUbmnþ f (2a x ) f (x ) 2b edIm,IRsayfacMnuc I(2 , 1) Cap©itbMElgqøúHénRkab ceKRtuVvRsay [eXIjfa f (4 x ) f (x ) 2 .

0 0

0

0

179

f (4 x 0 ) (4 x 0 ) 1 x0 3

0 0

1 2 (4 x 0 )

1 1 x0 3 2 x0 2 x0

1 nig f (x ) x 1 2 x . eK)an f (4 x ) f (x ) x 3 2 1x x 1 2 1x b¤ f (4 x ) f (x ) 2 Bit . dUcenH cMnuc I(2 , 1) Cap©itbMElgqøúHénRkab c . g> sg;Rkab (c) tagGnuKmn_ f (x) kñúgtMruyGrtUnrm:al; (0, i , j)

0 0 0 0 0 0 0 0

y

1

x 0 1

180

lMhat;TI29 x eK[GnuKmn_ y f (x) x3( x2) 3 4 k> cUrKNnalImIt lim f (x) , lim f (x) nig lim f (x) rYcTajrksmIkar GasIumtUtTaMgGs;énRkab c tagGnuKmn_ f . x> KNnaedrIev f ' (x) rYcKUstaragGefrPaBénGnuKmn_ f . K> bgðajfabnÞat; x 2 CaGkS½qøúHénRkab c . X> cUrsg;Rkab (c) tagGnuKmn_ f kñúgtRmuyGrtUnrm:al; (0, i , j) .

2 2

x 1 x 3 x

dMeNaHRsay k> KNnalImIt lim f (x) , lim f (x) nig lim f (x) x eKman y f (x) x3( x2) 3 4 x eK)an lim f (x) lim x3( x2) 3 4 x nig lim f (x) lim x3( x2) 3 4

x 1 x 3 x

2

2

2

x 1

x 1

2

2

x 1

x 1

2

nig TajrksmIkarGasIumtUtTaMgGs; ³ edayeKman ³ lim f ( x ) nig lim f ( x ) naM[bnÞat; x 1CaGasIumtUtQr .

x 1 x 1

3( x 2) 2 lim f ( x ) lim 2 x 3 x 3 x 4 x 3 3( x 2) 2 lim f ( x ) lim 2 x 3 x 3 x 4 x 3

181

nig lim f (x) naM[bnÞat; x 3 CaGasIumtUtQr. lim f ( x ) 3 naM[bnÞat; y 3 CaGasIumtUtedk . x> KNnaedrIev f ' (x) x eKman f (x) x3( x2) 3 kMnt;elI D IR {1,3} 4 eK)an f ' (x) 6(x 2)(x (4xx 3)x33()2x 4)(x 2) 4

x 3 x 3

lim f ( x )

x

2

2

f

2

2

2

2

6( x 2)( x 2 4 x 3) 6( x 2)( x 2) 2 ( x 2 4 x 3) 2 6( x 2) 6( x 2) ( x 2 4 x 3) ( x 2) 2 2 ( x 2 4 x 3) 2 ( x 4 x 3) 2

12 dUcenH f ' (x ) (x 6 x4x 3) . KUstaragGefrPaBénGnuKmn_ f ³ ebI f ' (x) 0 eK)an 6x 12 0 naM[ x 2 . 2 cMeBaH x 2 eK)an f (2) 34(2 )3 0 ¬ CacMnucbrma ¦. 8

2 2

2

x

f ' (x) f (x)

1

2

3

182

K> bgðajfabnÞat; x 2 CaGkS½qøúHénRkab c tamrUbmnþ f (2a x ) f (x ) edIm,IRsayfabnÞat; x 2 CaGkS½qøúHénRkab (c) eyIgRKan;EtRsay[eXIjfa ³ f ( 4 x ) f ( x ) RKb; x D . x eKman f (x) x3( x2) 3 kMnt;elI D IR {1,3} . 4 eK)an f (4 x ) (4 x3()4 x4(4 2)x ) 3

0 0 0 0 0 f

2

2

f

2

0

0

2

0

0

f (4 x 0 )

3( 2 x 0 ) 2 16 8x 0 x 0 16 4 x 0 3

2

Bit . dUcenH bnÞat; x 2 CaGkS½qøúHénRkab c . X> sg;Rkab (c) tagGnuKmn_ f kñúgtRmuyGrtUnrm:al; (0, i , j)

3( x 0 2) 2 f (4 x 0 ) 2 f (x 0 ) x 0 4x 0 3

y

(C)

1

x 0 1

183

lMhat;TI30 eK[GnuKmn_ f x x 1 e kMnt;elI IR . k-KNnalImIténGnuKmn_ f x kalNa x nig x . x-bgðajfaExSekag c tagGnuKmn_ y f x manGasIumtUteRTtmYykalNa x . K-KUstaragGefrPaBénGnuKmn_ f x .

x

dMeNaHRsay k-KNnalImIt eK)an lim f x limx 1 e eRBaH limx 1 nig lim e 0 .

x x x x x x

x 1 lim f x limx 1 e x lim e x . x 1 x x x e x lim e x x 1 lim x 0 x e

eRBaH

x-bgðajfaExSekag c tagGnuKmn_ y f x manGasIumtUteRTtmYy³ eKman f x x 1 e eday lim e 0 naM[bnÞat; y x 1 CaGasIumtUteRTténRkab c.

x x x

184

K-KUstaragGefrPaBénGnuKmn_ f x eKman f x x 1 e naM[ f ' x 1 e ebI f ' 0 0 eK)an e 1 naM[ x 0 ehIy f 0 0 1 e 1 1 0 . eyIgGacKUstaragGefrPaBdUcxageRkam ³

x x 0

x

x

f ' (x) f (x)

0

y

4

3

2

1

x -5 -1 -4 -3 -2 -1 0 1 2 3 4 5

-2

-3

-4

-5

-6

185

lMhat;TI31 eK[GnuKmn_ f kMnt;elI 0, eday f x x 1 ln x k-KNnalImIt lim f x nig lim f x . x-KUstaragGefrPaBénGnuKmn_ f x . K-cUrsg;Rkab ctMnagGnuKmn_ y f x kñúgtMruyGrtUNrma:l; X-KNnaRkLaépÞxN½ÐedayExSekag c CamYyGkS½Gab;sIus kñúgcenøaH 1, e .

x 0 x

dMeNaHRsay k-KNnalImIt lim f x nig lim f x lim f x limx 1 ln x eRBaH lim ln x .

x 0 x x 0 x 0 x 0

eRBaH . x-KUstaragGefrPaBénGnuKmn_ f x eKman f x x 1 ln x manEdnkMnt; D 0, eK)an f ' x 1 1 x x 1 ebI f ' x x x 1 0 naM[ x 1 x nig f 1 0 .

186

1 ln x lim f x lim x 1 ln x lim x 1 x x x x x 1 ln x lim 1 1 x x x

x

f ' x f x

0

1

K-sg;Rkab ctMnagGnuKmn_ y f x kñúgtMruyGrtUNrma:l;

y 2

1

3 x 0 1 2

-1

X-KNnaRkLaépÞ tag S CaépÞxN½ÐedayExSekag c CamYyGkS½Gab;sIuskñúgcenøaH 1, e eK)an S x 1 ln x .dx x 1.dx ln x.dx

e e e 1 1 1

2 x2 e2 1 e2 1 e 1 x 1.dx 2 x 2 e 2 1 2 e 2 2 1 1 e

e

e

ln x.dx x ln x x 1 e ln e e 1 ln 1 1 e e 1 1

1

e

dUcenH S e 1

2

2

e 2 2e 1 1 2

¬ ÉktaépÞ ¦ .

187

lMhat;TI32 eK[GnuKmn_ f x xe 1 manRkabtMnag c . k-KNnalImIt lim f x nig lim f x rYcbBa¢ak;smIkar GasIumtUtedkmYyénRkab c . x-KUstaragGefrPaBénGnuKmn_ f x . K-cUrsg;Rkab ckñúgtMruyGrtUNrm:al; 0, i , j . X-KNnaRkLaépÞ S énbøg;xN½ÐedayRkab CamYyGkS½Gab;suIs kñúgcenøaH 1, , 1 rYcTajrk lim S .

2x x x

dMeNaHRsay k-KNnalImIt lim f x nig lim f x ³ eK)an lim f x lim xe 1 limx 1e

x x x x 2x x

2 x

eRBaH nig eRBaH

limx 1 x 2 x lim e x x 1 lim f x lim 2 x limx 1e 2 x 0 x x e x limx 1 x 2 x lim e 0 x

naM[bnÞat; y 0 CasmIkarGasIumtUtedkmYyénRkab c .

188

x-KUstaragGefrPaBénGnuKmn_ f x eKman f x xe 1 x 1.e kMnt;elI IR eK)an f ' x e 2x 1.e

2 x 2x 2 x 2 x

2 x 1.e 2 x

ebI f ' x 2x 1.e 0 naM[ x 1 2 e cMeBaH x 1 naM[ f 1 2 . 2 2 taragGefrPaB

2 x

x

f ' x f x

1 2

K-sg;Rkab ckñúgtMruyGrtUNrm:al;

y 3 2

0, i , j

1

x -2 -1 0 1 2 3

-1

189

X-KNnaRkLaépÞ S ³ eyIg)an S f x .dx x 1.e

1 1

2 x

.dx

tag

du dx 1 v .e 2 x 2 1 2 x 1 2 x S x 1.e e .dx 2 1 2 1 1 1 1.e 2 e 2 x 1 2 4 1 1 1.e 2 e 2 e 2 2 4 2 1 2 1 2 e 2 e e e 2 2 4 4 e2 1 2 3.e 2 4 4 u x 1 dv e 2 x .dx

naM[

dUcenH nig ¬eRBaH lim 1 2 3.e 4

e2 1 S 2 3.e 2 4 4 e2 e2 1 2 lim S (2 3).e lim 4 4 4

2

¬ÉktþaépÞ¦ . .

0

¦.

190

lMhat;TI33 eK[GnuKmn_ f x 1 x .e 1 kMnt;elI IR . k> KNnaedrIev f 'x rYcKUstaragGefrPaBén f . TajbBa¢ak;sBaØaénGnuKn_ f x . x> eK[ g CaGnuKmn_ kMnt;elI IR eday gx 2 x .e 2 x . cUrKNnalImIt lim gx nig lim gx . K> KNnaedrIev g'x rYcbBa¢ak;sBaØarbs; g'x . KUstaragGefrPaBén gx . X> RsaybBa¢ak;fabnÞat; d : y 2 x CaGasIumtUteRTténRkab C tagGnuKmn_ g kalNa x . swkSaTItaMgeFobrvagExSekag C nigbnÞat; (d) . g>sresrsmIkarbnÞat; T b:HnwgExSekag C ehIyRsbnwgbnÞat; d c> kMnt;kUGredaencMnucrbt; I rbs;ExSekag C . q> cUrsg;Rkab C bnÞat; T nig d kñúgtMruyGrtUNrm:al; 0, i , j .

x x x x

dMeNaHRsay k> KNnaedrIev f 'x rYcKUstaragGefrPaBén f eKman f 'x 1 x '.e e '.1 x

x x

191

e x e x 1 x

e x e x x.e x

x.e x

ebI f 'x x.e 0 naMeGay x 0 . cMeBaH x 0 eK)an f 0 1 0.e 1 0 . KNnalImIt³

0

x

lim f x lim1 x .e x 1 1 x x

nig lim f x lim1 x .e

x x

x

1 0

x

f ' x f x

TajbBa¢ak;sBaØaénGnuKn_ f x ³ tamtaragxagelIeKTaj)an x IR : f x 0 . x KNnalImIt lim gx nig lim gx ³ eK)an lim gx lim2 x .e 2 x lim2 x eRBaH lime 0

x x x x x x x

x

x

eK)an lim gx lim2 x .e

x

x

2 x

192

eRBaH

lim2 x x x lime x

K> KNnaedrIev g'x rYcbBa¢ak;sBaØarbs; g'x ³ eKman gx 2 x .e 2 x 2 x e 1 eK)an g'x 2 x 'e 1 e 1'2 x

x x x x

e x 1 e x .2 x

e x 1 2e x x.e x e x x.e x 1 1 x .e x 1

dUcenH g'x 1 x .e 1 müa:geToteday g'x 1 x .e ehIyeKman x IR : f x 0 dUcenH x IR : g' x 0 . KUstaragGefrPaBén gx ³

x

g ' x g x

x

x

1 f x

0

cMeBaH x 0 naMeGay g0 4

193

X> RsaybBa¢ak;fabnÞat; d : y 2 x CaGasIumtUteRTténRkab C C : g x 2 x .e 2 x eKman d : y 2 x eK)an gx y 2 x .e edayeKman limgx y lim2 x .e 0 dUcenH bnÞat; d : y 2 x CaGasIumtUteRTténRkab C . sikSaTItaMgeFobrvagExSekag C nigbnÞat; (d) ³ eKman gx y 2 x .e mansBaØadUc 2 x eRBaH x IR : e 0 taragsBaØaén gx y 2 x .e

x x x x x x x x

x

g x y

2

-cMeBaH x ,2 ExSekag C enABIelIbnÞat; d . -cMeBaH x 2 ExSekag C RbsBVbnÞat; d Rtg;cMnuc A2,0 . -cMeBaH x 2, ExSekag C enABIeRkambnÞat; d . g> sresrsmIkarbnÞat; T b:HnwgExSekag C ehIyRsbnwgbnÞat; d ³ tag M x , y CacMnucb:HrvagbnÞat; T CamYy C tamrUbmnþ (T) : y y y' .x x

0 0 0 0 0 0

194

eday T //d : y 2 x naM[ y' 1 Et y' g'x 1 x e 1 eKTaj)an 1 x e 1 1 naM[ x 1 ehIy y gx e 1 eK)an T : y e 1 1.x 1 dUcenH T : y x e 2 . c> kMnt;kUGredaencMnucrbt; I rbs;ExSekag C ³ eKman g'x 1 x .e 1 f x eK)an g' 'x f 'x x.e manb¤s x 0 . cMeBaH x 0 eK)an g(0) 4 . taragsikSasBaØaén g' 'x x.e

0 x0 0 0 0 x0 0 0 0 0 x x x

x

g ' ' x g x

0

edayRtg;cMnuc x 0 kenSam g' 'x bþÚrsBaØaBI eTA naM[ I0,4 CacMnucrbt;énRkab .

195

q> sg;Rkab C bnÞat; T nig d kñúgtMruyGrtUNrm:al; ³

y 6 5

4

3

2

1

x -3 -1 -2 -1 0 1 2 3 4 5

-2

196

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