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LEARNING OUTCOMES:

1. Use the dividing out technique to evaluate limits of functions.

2. Use the rationalizing technique to evaluate the limits of functions.

3. Use technology to approximate the limits of functions graphically and numerically.

4. Evaluate one-sided limits of functions 5. Evaluate limits of difference quotients from calculus.

1. Use the dividing out technique to evaluate limits of functions.

2. Use the rationalizing technique to evaluate the limits of functions.

3. Use technology to approximate the limits of functions graphically and numerically.

4. Evaluate one-sided limits of functions 5. Evaluate limits of difference quotients from calculus.

1.
Techniques for

11.2 Evaluating Limits

Copyright © Cengage Learning. All rights reserved.

11.2 Evaluating Limits

Copyright © Cengage Learning. All rights reserved.

2.
What You Should Learn

• Use the dividing out technique to evaluate limits

of functions

• Use the rationalizing technique to evaluate

limits of functions

• Use technology to approximate limits of

functions graphically and numerically

2

• Use the dividing out technique to evaluate limits

of functions

• Use the rationalizing technique to evaluate

limits of functions

• Use technology to approximate limits of

functions graphically and numerically

2

3.
What You Should Learn

• Evaluate one-sided limits of functions

• Evaluate limits of difference quotients from

calculus

3

• Evaluate one-sided limits of functions

• Evaluate limits of difference quotients from

calculus

3

4.
Dividing Out Technique

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5.
Dividing Out Technique

We have studied several types of functions whose limits

can be evaluated by direct substitution.

In this section, you will study several techniques for

evaluating limits of functions for which direct substitution

Suppose you were asked to find the following limit.

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We have studied several types of functions whose limits

can be evaluated by direct substitution.

In this section, you will study several techniques for

evaluating limits of functions for which direct substitution

Suppose you were asked to find the following limit.

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6.
Dividing Out Technique

Direct substitution fails because –3 is a zero of the

denominator. By using a table, however, it appears that the

limit of the function as x approaches –3 is –5.

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Direct substitution fails because –3 is a zero of the

denominator. By using a table, however, it appears that the

limit of the function as x approaches –3 is –5.

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7.
Example 1 – Dividing Out Technique

Find the limit.

Begin by factoring the numerator and dividing out any

common factors.

Factor numerator.

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Find the limit.

Begin by factoring the numerator and dividing out any

common factors.

Factor numerator.

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8.
Example 1 – Solution cont’d

Divide out common

factor.

(x – 2) Simplify.

= –3 – 2 Direct substitution

= –5 Simplify.

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Divide out common

factor.

(x – 2) Simplify.

= –3 – 2 Direct substitution

= –5 Simplify.

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9.
Dividing Out Technique

This procedure for evaluating a limit is called the dividing

out technique.

The validity of this technique stems from the fact that when

two functions agree at all but a single number c, they must

have identical limit behavior at x = c.

In Example 1, the functions given by

f (x) and g (x) = x – 2

agree at all values of x other than x = –3.

So, you can use g (x) to find the limit of f (x).

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This procedure for evaluating a limit is called the dividing

out technique.

The validity of this technique stems from the fact that when

two functions agree at all but a single number c, they must

have identical limit behavior at x = c.

In Example 1, the functions given by

f (x) and g (x) = x – 2

agree at all values of x other than x = –3.

So, you can use g (x) to find the limit of f (x).

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10.
Dividing Out Technique

The dividing out technique should be applied only when

direct substitution produces 0 in both the numerator and

the denominator.

An expression such as has no meaning as a real

It is called an indeterminate form because you cannot,

from the form alone, determine the limit.

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The dividing out technique should be applied only when

direct substitution produces 0 in both the numerator and

the denominator.

An expression such as has no meaning as a real

It is called an indeterminate form because you cannot,

from the form alone, determine the limit.

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11.
Dividing Out Technique

When you try to evaluate a limit of a rational function by

direct substitution and encounter this form, you can

conclude that the numerator and denominator must have a

common factor.

After factoring and dividing out, you should try direct

substitution again.

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When you try to evaluate a limit of a rational function by

direct substitution and encounter this form, you can

conclude that the numerator and denominator must have a

common factor.

After factoring and dividing out, you should try direct

substitution again.

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12.
Rationalizing Technique

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13.
Rationalizing Technique

Another way to find the limits of some functions is first to

rationalize the numerator of the function. This is called the

rationalizing technique.

We have known that rationalizing the numerator means

multiplying the numerator and denominator by the

conjugate of the numerator.

For instance, the conjugate of + 4 is – 4.

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Another way to find the limits of some functions is first to

rationalize the numerator of the function. This is called the

rationalizing technique.

We have known that rationalizing the numerator means

multiplying the numerator and denominator by the

conjugate of the numerator.

For instance, the conjugate of + 4 is – 4.

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14.
Example 3 – Rationalizing Technique

Find the limit.

By direct substitution, you obtain the indeterminate form .

Indeterminate form

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Find the limit.

By direct substitution, you obtain the indeterminate form .

Indeterminate form

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15.
Example 3 – Solution cont’d

In this case, you can rewrite the fraction by rationalizing the

Multiply.

Simplify.

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In this case, you can rewrite the fraction by rationalizing the

Multiply.

Simplify.

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16.
Example 3 – Solution cont’d

Divide out common factor.

Simplify.

,x≠0

Now you can evaluate the limit by direct substitution.

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Divide out common factor.

Simplify.

,x≠0

Now you can evaluate the limit by direct substitution.

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17.
Example 3 – Solution cont’d

You can reinforce your conclusion that the limit is by

constructing a table, as shown below, or by sketching a

graph, as shown in Figure 11.12.

Figure 11.12

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You can reinforce your conclusion that the limit is by

constructing a table, as shown below, or by sketching a

graph, as shown in Figure 11.12.

Figure 11.12

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18.
One-Sided Limits

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19.
One-Sided Limits

The limit of f (x) as x c does not exist when the function

f (x) approaches a different number from the left side of c

than it approaches from the right side of c.

This type of behavior can be described more concisely with

the concept of a one-sided limit.

f (x) = L1 or f (x) L1 as x c– Limit from the left

Limit from the right

f (x) = L2 or f (x) L2 as x c+

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The limit of f (x) as x c does not exist when the function

f (x) approaches a different number from the left side of c

than it approaches from the right side of c.

This type of behavior can be described more concisely with

the concept of a one-sided limit.

f (x) = L1 or f (x) L1 as x c– Limit from the left

Limit from the right

f (x) = L2 or f (x) L2 as x c+

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20.
Example 6 – Evaluating One-Sided Limits

Find the limit as x 0 from the left and the limit as x 0

from the right for

f (x) = .

From the graph of f, shown in

Figure 11.15, you can see

that f (x) = –2 for all x < 0.

Figure 11.15

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Find the limit as x 0 from the left and the limit as x 0

from the right for

f (x) = .

From the graph of f, shown in

Figure 11.15, you can see

that f (x) = –2 for all x < 0.

Figure 11.15

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21.
Example 6 – Solution cont’d

So, the limit from the left is

Limit from the left

= –2.

Because f (x) = 2 for all x > 0, the limit from the right is

= 2. Limit from the right

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So, the limit from the left is

Limit from the left

= –2.

Because f (x) = 2 for all x > 0, the limit from the right is

= 2. Limit from the right

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22.
One-Sided Limits

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23.
A Limit from Calculus

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24.
A Limit from Calculus

A Limit from Calculus In the next section, you will study an

important type of limit from calculus—the limit of a

difference quotient.

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A Limit from Calculus In the next section, you will study an

important type of limit from calculus—the limit of a

difference quotient.

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25.
Example 9 – Evaluating a Limit from Calculus

For the function given by f (x) = x2 – 1, find

Direct substitution produces an indeterminate form.

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For the function given by f (x) = x2 – 1, find

Direct substitution produces an indeterminate form.

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26.
Example 9 – Solution cont’d

By factoring and dividing out, you obtain the following.

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By factoring and dividing out, you obtain the following.

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27.
Example 9 – Solution cont’d

= (6 + h)

=6+0

=6

So, the limit is 6.

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= (6 + h)

=6+0

=6

So, the limit is 6.

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