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OBJECTIVES:

1. Use the definition of limit to estimate limits.

2. Determine whether limits of functions exist.

3. Use properties of limits and direct substitution to evaluate limits.

1. Use the definition of limit to estimate limits.

2. Determine whether limits of functions exist.

3. Use properties of limits and direct substitution to evaluate limits.

1.
12 Limits and an Introduction

to Calculus

Copyright © Cengage Learning. All rights reserved.

to Calculus

Copyright © Cengage Learning. All rights reserved.

2.
12.1 Introduction to Limits

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

3.
Use the definition of limit to estimate limits.

Determine whether limits of functions exist.

Use properties of limits and direct substitution to

evaluate limits.

3

Determine whether limits of functions exist.

Use properties of limits and direct substitution to

evaluate limits.

3

4.
The Limit Concept

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5.
The Limit Concept

The notion of a limit is a fundamental concept of calculus.

In this chapter, you will learn how to evaluate limits and

how to use them in the two basic problems of calculus: the

tangent line problem and the area problem.

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The notion of a limit is a fundamental concept of calculus.

In this chapter, you will learn how to evaluate limits and

how to use them in the two basic problems of calculus: the

tangent line problem and the area problem.

5

6.
Example 1 – Finding a Rectangle of Maximum Area

Find the dimensions of a rectangle that has a perimeter of

24 inches and a maximum area.

Let w represent the width of the rectangle and let l

represent the length of the rectangle. Because

2w + 2l = 24 Perimeter is 24.

it follows that l = 12 – w,

as shown in the figure.

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Find the dimensions of a rectangle that has a perimeter of

24 inches and a maximum area.

Let w represent the width of the rectangle and let l

represent the length of the rectangle. Because

2w + 2l = 24 Perimeter is 24.

it follows that l = 12 – w,

as shown in the figure.

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7.
Example 1 – Solution cont’d

So, the area of the rectangle is

A = lw Formula for area

= (12 – w)w Substitute 12 – w for l.

= 12w – w2. Simplify.

Using this model for area, experiment with different values

of w to see how to obtain the maximum area.

7

So, the area of the rectangle is

A = lw Formula for area

= (12 – w)w Substitute 12 – w for l.

= 12w – w2. Simplify.

Using this model for area, experiment with different values

of w to see how to obtain the maximum area.

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8.
Example 1 – Solution cont’d

After trying several values, it appears that the maximum

area occurs when w = 6, as shown in the table.

In limit terminology, you can say that “the limit of A as w

approaches 6 is 36.” This is written as

8

After trying several values, it appears that the maximum

area occurs when w = 6, as shown in the table.

In limit terminology, you can say that “the limit of A as w

approaches 6 is 36.” This is written as

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9.
Definition of Limit

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10.
Definition of Limit

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11.
Example 2 – Estimating a Limit Numerically

Use a table to estimate numerically the limit: .

Let f (x) = 3x – 2.

Then construct a table that shows values of f (x) for two

of x-values—one set that approaches 2 from the left and

one that approaches 2 from the right.

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Use a table to estimate numerically the limit: .

Let f (x) = 3x – 2.

Then construct a table that shows values of f (x) for two

of x-values—one set that approaches 2 from the left and

one that approaches 2 from the right.

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12.
Example 2 – Solution cont’d

From the table, it appears that the closer x gets to 2, the

closer f (x) gets to 4. So, you can estimate the limit to be 4.

Figure 12.1 illustrates this conclusion.

Figure 12.1

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From the table, it appears that the closer x gets to 2, the

closer f (x) gets to 4. So, you can estimate the limit to be 4.

Figure 12.1 illustrates this conclusion.

Figure 12.1

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13.
Limits That Fail to Exist

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14.
Limits That Fail to Exist

Next, you will examine some functions for which limits do

not exist.

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Next, you will examine some functions for which limits do

not exist.

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15.
Example 6 – Comparing Left and Right Behavior

Show that the limit does not exist.

Consider the graph of f (x) = | x |/x.

From Figure 12.4, you can see that

for positive x-values

and for negative x-values

Figure 12.4

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Show that the limit does not exist.

Consider the graph of f (x) = | x |/x.

From Figure 12.4, you can see that

for positive x-values

and for negative x-values

Figure 12.4

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16.
Example 6 – Solution cont’d

This means that no matter how close x gets to 0, there will

be both positive and negative x-values that yield f (x) = 1

and f (x) = –1.

This implies that the limit does not exist.

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This means that no matter how close x gets to 0, there will

be both positive and negative x-values that yield f (x) = 1

and f (x) = –1.

This implies that the limit does not exist.

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17.
Limits That Fail to Exist

Following are the three most common types of behavior

associated with the nonexistence of a limit.

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Following are the three most common types of behavior

associated with the nonexistence of a limit.

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18.
Properties of Limits and

Direct Substitution

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Direct Substitution

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19.
Properties of Limits and Direct Substitution

You have seen that sometimes the limit of f (x) as x c is

simply f (c), as shown in Example 2. In such cases, the limit

can be evaluated by direct substitution.

That is,

Substitute c for x.

There are many “well-behaved” functions, such as

polynomial functions and rational functions with nonzero

denominators, that have this property.

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You have seen that sometimes the limit of f (x) as x c is

simply f (c), as shown in Example 2. In such cases, the limit

can be evaluated by direct substitution.

That is,

Substitute c for x.

There are many “well-behaved” functions, such as

polynomial functions and rational functions with nonzero

denominators, that have this property.

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20.
Properties of Limits and Direct Substitution

The following list includes some basic limits.

This list can also include trigonometric functions. For

and

20

The following list includes some basic limits.

This list can also include trigonometric functions. For

and

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21.
Properties of Limits and Direct Substitution

By combining the basic limits with the following operations,

you can find limits for a wide variety of functions.

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By combining the basic limits with the following operations,

you can find limits for a wide variety of functions.

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22.
Example 9 – Direct Substitution and Properties of Limits

Find each limit.

a. b. c.

d. e. f.

Use the properties of limits and direct substitution to

evaluate each limit.

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Find each limit.

a. b. c.

d. e. f.

Use the properties of limits and direct substitution to

evaluate each limit.

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23.
Example 9 – Solution cont’d

b. Property 1

c. Property 4

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b. Property 1

c. Property 4

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24.
Example 9 – Solution cont’d

e. Property 3

f. Properties 2 and 5

24

e. Property 3

f. Properties 2 and 5

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25.
Properties of Limits and Direct Substitution

Example 9 shows algebraic solutions. To verify the limit in

Example 9(a) numerically, for instance, create a table that

shows values of x2 for two sets of x-values—one set that

approaches 4 from the left and one that approaches 4 from

the right, as shown below.

25

Example 9 shows algebraic solutions. To verify the limit in

Example 9(a) numerically, for instance, create a table that

shows values of x2 for two sets of x-values—one set that

approaches 4 from the left and one that approaches 4 from

the right, as shown below.

25

26.
Properties of Limits and Direct Substitution

From the table, you can see that the limit as x approaches

4 is 16. To verify the limit graphically, sketch the graph

of y = x2. From the graph shown in Figure 12.7, you can

determine that the limit as x approaches 4 is 16.

Figure 12.7

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From the table, you can see that the limit as x approaches

4 is 16. To verify the limit graphically, sketch the graph

of y = x2. From the graph shown in Figure 12.7, you can

determine that the limit as x approaches 4 is 16.

Figure 12.7

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27.
Properties of Limits and Direct Substitution

The following summarizes the results of using direct

substitution to evaluate limits of polynomial and rational

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The following summarizes the results of using direct

substitution to evaluate limits of polynomial and rational

27