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Chapter 12

Section 12.1

Limits and an Introduction to Calculus

Course Number Instructor Date

Introduction to Limits

Objective: In this lesson you learned how to estimate limits and use properties and operations of limits.

I. The Limit Concept and Definition of Limit (Pages 806-808) Define limit. aa a aaa aaaaaaa aaaaaaaaaaa aaaaa aa a aaaaaa aaaaaa a aa a aaaaaaaaaa a aaaa aaaaaa aaaaa aaa aaaaa aa a aaa aa a aaaaaaaaaa a aa aa aaaa aaaaaaa aa aaa a aaa a aa aaa aa Describe how to estimate the limit lim x 2 + 4x + 4 numerically. x - 2 x+2

What you should learn How to use the definition of a limit to estimate limits

aaa a aaa a aaa a aa a aaaaa a aaa aaaa aaaaaaaaa a aaaaa aaaa aaaaa aaaaaa aa a aaa aaaa a aa aaaaa aa a aa aaa aaa aaaaa aa aaaa aaa a aaaaaaaaa aaaaa aa aaa aaaaa aa a aaa aa a aaaaaaaaaa a aa aaaa aa aa aaaaaaaa aa aaa aaaaaa The existence or nonexistence of f (x) when x = c has no bearing on the existence of . . . aaa aaaaa aa a aaa aa a aaaaaaaaaa

II. Limits That Fail to Exist (Pages 809-810) The limit of f (x) as x c does not exist if any of the following conditions is true: 1. a aaa aaaaaaaaaa a aaaaaaaaa aaaaaa aaaa aaa aaaaa aaaa aa a aaaa aaaa aaa aaaa aaaa aa aa 2. a aaa aaaaaaaaa aa aaaaaaaaa aaaaaaa aaaaa aa a aaaaaaaaaa 3. a aaa aaaaaaaaaa aaaaaaa aaa aaaaa aaaaaa aa a aaaaaaaaaa

What you should learn How to decide whether limits of functions exist

Give an example of a limit that does not exist. aaaaaaa aaaa aaaaa

Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Third Edition Student Success Organizer Copyright © Houghton Mifflin Company. All rights reserved.

189

190 III. Properties of Limits (Pages 811-812)

Chapter 12

Limits and an Introduction to Calculus

Let b and c be real numbers and let n be a positive integer. Complete each of the following properties of limits. 1. lim b =

x c

What you should learn How to use properties and operations of limits to find limits

a a aa

a

a a

2. lim x =

x c

3. lim x n =

x c

a a

4. lim n x =

xc

a a a aaa a aaaa aaa a a a

Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits. lim f ( x ) = L and lim g ( x) = K

x c xc

Complete each of the following statements about operations with limits. 1. Scalar multiple: 2. Sum or difference: 3. Product:

x c

lim [b f ( x)] = lim [ f ( x) ± g ( x )] = lim [ f ( x) g ( x )] = f ( x) = g ( x)

aa

a aaa aa a a

x c

x c

4. Quotient:

x c

lim

aaaa aaaaaaaa a a a

a

5. Power:

x c

lim [ f ( x)] n = 4 - x2 . x 2 x

aa

a

Example 1: Find the limit: lim a Homework Assignment Page(s) Exercises

Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Third Edition Student Success Organizer Copyright © Houghton Mifflin Company. All rights reserved.

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