Section 12.4 Limits at Infinity and Limits of Sequences 883

Limits at Infinity and Horizontal AsymptotesAs pointed out at the beginning of this chapter, there are two basic problems incalculus: finding tangent lines and finding the area of a region. In Section 12.3,you saw how limits can be used to solve the tangent line problem. In this sectionand the next, you will see how a different type of limit, a limit at infinity, can beused to solve the area problem. To get an idea of what is meant by a limit at infinity, consider the function given by

The graph of is shown in Figure 12.30. From earlier work, you know that is a horizontal asymptote of the graph of this function. Using limit notation, thiscan be written as follows.

Horizontal asymptote to the left

Horizontal asymptote to the right

These limits mean that the value of gets arbitrarily close to as decreasesor increases without bound.

FIGURE 12.30

−1−2−3 1 2 3

−2

−3

1

2

3

x

y

x + 12x

f (x) =

y = 12

x12f �x�

limx→�

f �x� �1

2

limx→��

f �x� �1

2

y �12f

f �x� �x � 1

2x.

What you should learn• Evaluate limits of functions

at infinity.

• Find limits of sequences.

Why you should learn itFinding limits at infinity is useful in many types of real-lifeapplications. For instance, inExercise 52 on page 891, you are asked to find a limit at infinityto determine the number of military reserve personnel in thefuture.

Limits at Infinity and Limits of Sequences

© Karen Kasmauski/Corbis

12.4

Definition of Limits at InfinityIf is a function and and are real numbers, the statements

Limit as approaches

and

Limit as approaches

denote the limits at infinity. The first statement is read “the limit of asapproaches is ” and the second is read “the limit of as

approaches is ”L2.�xf �x�L1,��x

f �x�

�x limx→�

f �x� � L2

��x limx→��

f �x� � L1

L2L1f

332522_1204.qxd 12/13/05 1:06 PM Page 883

To help evaluate limits at infinity, you can use the following definition.

Limits at infinity share many of the properties of limits listed in Section12.1. Some of these properties are demonstrated in the next example.

884 Chapter 12 Limits and an Introduction to Calculus

Use a graphing utility to graphthe two functions given by

and

in the same viewing window.Why doesn’t appear to theleft of the -axis? How doesthis relate to the statement atthe right about the infinite limit

limx→��

1

xr?

yy1

y2 �1

3�xy1 �

1�x

Exploration

Limits at InfinityIf is a positive real number, then

Limit toward the right

Furthermore, if is defined when then

Limit toward the left limx→��

1

xr� 0.

x < 0,xr

limx→�

1

xr� 0.

r

Evaluating a Limit at Infinity

Find the limit.

limx→�

�4 �3x2�

Example 1

Algebraic SolutionUse the properties of limits listed in Section 12.1.

So, the limit of as approaches is 4.

Now try Exercise 5.

�xf �x� � 4 �3

x2

� 4

� 4 � 3�0�

� limx→�

4 � 3�limx→�

1

x2�

limx→�

�4 �3x2� � lim

x→� 4 � lim

x→� 3x2

Graphical SolutionUse a graphing utility to graph Then use thetrace feature to determine that as gets larger and larger, getscloser and closer to 4, as shown in Figure 12.31. Note that theline is a horizontal asymptote to the right.

FIGURE 12.31

−1

−20 120

5y = 4

y = 4 − 3x2

y � 4

yxy � 4 � 3�x2.

In Figure 12.31, it appears that the line is also a horizontal asymptoteto the left. You can verify this by showing that

The graph of a rational function need not have a horizontal asymptote. If it does,however, its left and right horizontal asymptotes must be the same.

When evaluating limits at infinity for more complicated rational functions,divide the numerator and denominator by the highest-powered term in the denom-inator. This enables you to evaluate each limit using the limits at infinity at thetop of the page.

limx→��

�4 �3x2� � 4.

y � 4

332522_1204.qxd 12/13/05 1:06 PM Page 884

Comparing Limits at Infinity

Find the limit as approaches for each function.

a. b. c.

SolutionIn each case, begin by dividing both the numerator and denominator by thehighest-powered term in the denominator.

a.

b.

c.

In this case, you can conclude that the limit does not exist because thenumerator decreases without bound as the denominator approaches 3.

Now try Exercise 13.

In Example 2, observe that when the degree of the numerator is less than thedegree of the denominator, as in part (a), the limit is 0. When the degrees of thenumerator and denominator are equal, as in part (b), the limit is the ratio of thecoefficients of the highest-powered terms. When the degree of the numerator isgreater than the degree of the denominator, as in part (c), the limit does not exist.

This result seems reasonable when you realize that for large values of the highest-powered term of a polynomial is the most “influential” term. That is, apolynomial tends to behave as its highest-powered term behaves as approachespositive or negative infinity.

x

x,

limx→�

�2x3 � 3

3x2 � 1� lim

x→�

�2x �3

x2

3 �1

x2

� �2

3

��2 � 0

3 � 0

limx→�

�2x2 � 3

3x2 � 1� lim

x→�

�2 �3

x2

3 �1

x2

� 0

��0 � 0

3 � 0

limx→�

�2x � 3

3x2 � 1� lim

x→�

�2

x�

3

x2

3 �1

x2

x2,

f �x� ��2x3 � 3

3x2 � 1f �x� �

�2x2 � 3

3x2 � 1f �x� �

�2x � 3

3x2 � 1

�x

Section 12.4 Limits at Infinity and Limits of Sequences 885

Example 2

Use a graphing utility to complete the table below to verify that

Make a conjecture about

limx→0

1x.

limx→�

1x

� 0.

Exploration

x

1x

105104103

x

1x

102101100

Have students use these observationsfrom Example 2 to predict the followinglimits.

a.

b.

c.

Then ask several students to verify thepredictions algebraically, several otherstudents to verify the predictionsnumerically, and several more studentsto verify the predictions graphically.Lead a discussion comparing the results.

limx→�

�6x2 � 1

3x2 � x � 2

limx→�

4x3 � 5x

8x4 � 3x2 � 2

limx→�

5x�x � 3�2x

332522_1204.qxd 12/13/05 1:06 PM Page 885

Finding the Average Cost

You are manufacturing greeting cards that cost $0.50 per card to produce. Yourinitial investment is $5000, which implies that the total cost of producing cards is given by The average cost per card is given by

Find the average cost per card when (a) (b) and (c) (d) What is the limit of as approaches infinity?

Solutiona. When the average cost per card is

b. When the average cost per card is

c. When the average cost per card is

d. As approaches infinity, the limit of is

The graph of is shown in Figure 12.32.

Now try Exercise 49.

C

x → �limx→�

0.50x � 5000

x� $0.50.

Cx

� $0.55.

x � 100,000 C �0.50�100,000� � 5000

100,000

x � 100,000,

� $1.00.

x � 10,000 C �0.50�10,000� � 5000

10,000

x � 10,000,

� $5.50.

x � 1000 C �0.50�1000� � 5000

1000

x � 1000,

xCx � 100,000.x � 10,000,x � 1000,

C �C

x�

0.50x � 5000

x.

CC � 0.50x � 5000.xC

886 Chapter 12 Limits and an Introduction to Calculus

Limits at Infinity for Rational FunctionsConsider the rational function where

and

The limit of as approaches positive or negative infinity is as follows.

If the limit does not exist.n > m,

limx→±�

f �x� � �0,an

bm

,

n < m

n � m

xf�x�

D�x� � bmxm � . . . � b0.N�x� � anxn � . . . � a0

f�x� � N�x��D�x�,

Example 3

x

C

Number of cards

Ave

rage

cos

t per

car

d(i

n do

llars

)

20,000 60,000 100,000

1

2

3

4

5

6

Average Cost

y = 0.5

0.50x + 5000x

Cx

C = =

As the average cost per cardapproaches $0.50.FIGURE 12.32

x → �,

Consider asking your students to identify the practical interpretation of the limit in part (d) of Example 3.

332522_1204.qxd 12/13/05 1:06 PM Page 886

Limits of SequencesLimits of sequences have many of the same properties as limits of functions. Forinstance, consider the sequence whose th term is

As increases without bound, the terms of this sequence get closer and closer to 0, and the sequence is said to converge to 0. Using limit notation, youcan write

The following relationship shows how limits of functions of can be used toevaluate the limit of a sequence.

A sequence that does not converge is said to diverge. For instance, the termsof the sequence oscillate between 1 and Therefore, thesequence diverges because it does not approach a unique number.

Finding the Limit of a Sequence

Find the limit of each sequence. (Assume begins with 1.)

a.

b.

c.

Solution

a.

b.

c.

Now try Exercise 33.

3

4,

9

16,

19

36,

33

64,

51

100,

73

144, . . . →

1

2lim

n→� 2n2 � 1

4n2 �12

3

5,

5

8,

7

13,

9

20,

11

29,

13

40, . . . → 0lim

n→�

2n � 1

n2 � 4� 0

3

5,

5

6,

7

7,

9

8,

11

9,

13

10, . . . → 2lim

n→�

2n � 1

n � 4� 2

cn �2n2 � 1

4n2

bn �2n � 1

n2 � 4

an �2n � 1

n � 4

n

�1.1, �1, 1, �1, 1, . . .

x

limn→�

1

2n� 0.

n

1

2,

1

4,

1

8,

1

16,

1

32, . . .

an � 1�2n.n

Section 12.4 Limits at Infinity and Limits of Sequences 887

Limit of a SequenceLet be a function of a real variable such that

If is a sequence such that for every positive integer then

limn→�

an � L.

n,f �n� � an�an

limx→�

f �x� � L.

f

Example 4

You can use the definition oflimits at infinity for rationalfunctions on page 886 to verifythe limits of the sequences inExample 4.

Another sequence that diverges is

You might want your

students to discuss why this is true.

an �1

n�1�4.

There are a number of ways to usea graphing utility to generate theterms of a sequence. For instance,you can display the first 10 termsof the sequence

using the sequence feature or the table feature.

an �12n

Techno logy

332522_1204.qxd 12/13/05 1:06 PM Page 887

In the next section, you will encounter limits of sequences such as that shownin Example 5. A strategy for evaluating such limits is to begin by writing the thterm in standard rational function form. Then you can determine the limit by comparing the degrees of the numerator and denominator, as shown on page 886.

n

888 Chapter 12 Limits and an Introduction to Calculus

Finding the Limit of a Sequence

Find the limit of the sequence whose th term is

an �8

n3n�n � 1��2n � 1�6 �.

n

Example 5

Algebraic SolutionBegin by writing the th term in standard rational functionform—as the ratio of two polynomials.

Write original th term.

Multiply fractions.

Write in standard rational form.

From this form, you can see that the degree of the numerator isequal to the degree of the denominator. So, the limit of thesequence is the ratio of the coefficients of the highest-poweredterms.

Now try Exercise 43.

limn→�

8n3 � 12n2 � 4n

3n3�

8

3

�8n3 � 12n2 � 4n

3n3

�8�n��n � 1��2n � 1�

6n3

n an �8

n3n�n � 1��2n � 1�6 �

n

Numerical SolutionConstruct a table that shows the value of as becomes larger and larger, as shown below.

From the table, you can estimate that as approachesgets closer and closer to 2.667 � 8

3.an�,n

nan

1 8

10 3.08

100 2.707

1000 2.671

10,000 2.667

ann

W RITING ABOUT MATHEMATICS

Comparing Rates of Convergence In the table in Example 5 above, the value of approaches its limit of rather slowly. (The first term to be accurate to three

decimal places is ) Each of the following sequences converges to 0.Which converges the quickest? Which converges the slowest? Why? Write a shortparagraph discussing your conclusions.

a. b. c.

d. e. hn �2n

n!dn �

1

n!

cn �1

2nbn �

1

n2an �

1

n

a4801 � 2.667.

83an

332522_1204.qxd 12/13/05 1:06 PM Page 888

Section 12.4 Limits at Infinity and Limits of Sequences 889

In Exercises 1– 4, match the function with its graph, usinghorizontal asymptotes as aids. [The graphs are labeled (a),(b), (c), and (d).]

(a) (b)

(c) (d)

1. 2.

3. 4.

In Exercises 5–22, find the limit (if it exists). If the limit doesnot exist, explain why. Use a graphing utility to verify yourresult graphically.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15.

16.

17. 18.

19. 20.

21.

22.

In Exercises 23–28, use a graphing utility to graph thefunction and verify that the horizontal asymptotecorresponds to the limit at infinity.

23. 24.

25. 26.

27. 28.

Numerical and Graphical Analysis In Exercises 29–32,(a) complete the table and numerically estimate the limitas approaches infinity, and (b) use a graphing utility tograph the function and estimate the limit graphically.

29.

30.

31.

32. f �x� � 4�4x � �16x2 � x �f �x� � 3�2x � �4x2 � x �f �x� � 3x � �9x2 � 1

f �x� � x � �x2 � 2

x

y � 2 �1x

y � 1 �3

x2

y �2x � 1x2 � 1

y �2x

1 � x2

y �x2

x2 � 4y �

3x

1 � x

limx→�

x2x � 1

�3x2

�x � 3�2�

limt→�

� 13t2

�5t

t � 2�

limx→�

7 �2x2

�x � 3�2�limx→��

x�x � 1�2 � 4�

limx→�

2x2 � 6�x � 1�2lim

x→��

��x2 � 3��2 � x�2

limx→��

2x2 � 5x � 121 � 6x � 8x2

limt→�

1 � 2t � 6t2

5 � 3t � 4t2

limy→�

4y4

y 2 � 3lim

t→�

t 2

t � 3

limx→��

3x2 � 14x2 � 5

limx→��

3x2 � 41 � x2

limx→�

1 � 2xx � 2

limx→��

4x � 32x � 1

limx→�

1 � 6x1 � 5x

limx→�

3 � x

3 � x

limx→�

52x

limx→�

3

x2

f �x� � x �1

xf �x� � 4 �

1

x2

f �x� �x2

x2 � 1f �x� �

4x2

x2 � 1

−2−4 2 4 6−2

−4

−6

6

x

yy

−2−4 2 4 6−2

−4

−6

6

x

y

x−2−4 2 4 6

−4

−6

2

4

6

−1−2 1 2 3−1

−2

−3

2

3

x

y

Exercises 12.4

VOCABULARY CHECK: Fill in the blanks.

1. A________ at ________ can be used to solve the area problem in calculus.

2. A sequence that has a limit is said to ________.

3. A sequence that does not have a limit is said to ________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

x

f �x�

106105104103102101100

332522_1204.qxd 12/13/05 1:06 PM Page 889

In Exercises 33–42, write the first five terms of the sequenceand find the limit of the sequence (if it exists). If the limitdoes not exist, explain why. Assume n begins with 1.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

In Exercises 43– 46, find the limit of the sequence. Thenverify the limit numerically by using a graphing utility tocomplete the table.

43.

44.

45.

46.

47. Oxygen Level Suppose that measures the level of oxygen in a pond, where is the normal (unpolluted) level and the time is measured in weeks.When organic waste is dumped into the pond, and asthe waste material oxidizes, the level of oxygen in the pondis given by

(a) What is the limit of this function as approachesinfinity?

(b) Use a graphing utility to graph the function and verifythe result of part (a).

(c) Explain the meaning of the limit in the context of theproblem.

48. Typing Speed The average typing speed (in words perminute) for a student after weeks of lessons is given by

(a) What is the limit of this function as approachesinfinity?

(b) Use a graphing utility to graph the function and verifythe result of part (a).

(c) Explain the meaning of the limit in the context of theproblem.

49. Average Cost The cost function for a certain model ofpersonal digital assistant (PDA) is given by

where is measured in dollars andis the number of PDAs produced.

(a) Write a model for the average cost per unit produced.

(b) Find the average costs per unit when and

(c) Determine the limit of the average cost function as approaches infinity. Explain the meaning of the limit inthe context of the problem.

50. Average Cost The cost function for a company to recycletons of material is given by whereis measured in dollars.

(a) Write a model for the average cost per ton of materialrecycled.

(b) Find the average costs of recycling 100 tons of materialand 1000 tons of material.

(c) Determine the limit of the average cost function as approaches infinity. Explain the meaning of the limit inthe context of the problem.

51. Data Analysis: Social Security The table shows theaverage monthly Social Security benefits (in dollars) forretired workers aged 62 or over from 1997 to 2003.(Source: U.S. Social Security Administration)

A model for the data is given by

where represents the year, with corresponding to1997.

(a) Use a graphing utility to create a scatter plot of the dataand graph the model in the same viewing window. Howwell does the model fit the data?

t � 7t

B �199.0 � 999.46t

1.0 � 1.43t � 0.032t 2, 7 ≤ t ≤ 13

B

x

CC � 1.25x � 10,500,x

x

x � 1000.x � 100

xCC � 13.50x � 45,750,

t

S �100t 2

65 � t 2, t > 0.

tS

t

f �t� �t 2 � t � 1

t 2 � 1.

t � 0,tf �t� � 1f �t�

an �n�n � 1�

n2�

1

n4n�n � 1�2 �

2

an �16

n3n�n � 1��2n � 1�6 �

an �4

n�n �4

nn�n � 1�

2 ��

an �1

n�n �1

nn�n � 1�

2 ��

an ���1�n�1

n2an �

��1�n

n

an ��3n � 1�!�3n � 1�!

an ��n � 1�!

n!

an �4n2 � 1

2nan �

n2

3n � 2

an �4n � 1

n � 3an �

n2n � 1

an �n

n2 � 1an �

n � 1

n2 � 1

890 Chapter 12 Limits and an Introduction to Calculus

n

an

106105104103102101100

Year Benefit, B

1997 765

1998 780

1999 804

2000 844

2001 874

2002 895

2003 922

332522_1204.qxd 12/13/05 1:06 PM Page 890

Section 12.4 Limits at Infinity and Limits of Sequences 891

(b) Use the model to predict the average monthly benefitin 2006.

(c) Discuss why this model should not be used to predict theaverage monthly Social Security benefits in future years.

Synthesis

True or False? In Exercises 53–56, determine whether thestatement is true or false. Justify your answer.

53. Every rational function has a horizontal asymptote.

54. If increases without bound as approaches then thelimit of exists.

55. If a sequence converges, then it has a limit.

56. When the degrees of the numerator and denominator of arational function are equal, the limit does not exist.

57. Think About It Find the functions and such that bothand increase without bound as approaches but

58. Think About It Use a graphing utility to graph thefunction given by

How many horizontal asymptotes does the function appearto have? What are the horizontal asymptotes?

Exploration In Exercises 59–62, create a scatter plot of theterms of the sequence. Determine whether the sequenceconverges or diverges. If it converges, estimate its limit.

59. 60.

61. 62.

Skills Review

In Exercises 63 and 64, sketch the graphs of and eachtransformation on the same rectangular coordinate system.

63.

(a) (b)

(c) (d)

64.

(a) (b)

(c) (d)

In Exercises 65–68, divide using long division.

65.

66.

67.

68.

In Exercises 69–72, find all the real zeros of the polynomialfunction. Use a graphing utility to graph the function andverify that the real zeros are the -intercepts of the graphof the function.

69. 70.

71.

72.

In Exercises 73–76, find the sum.

73. 74.

75. 76. 8

k�0

3k2 � 1

10

k�115

4

i�05i2

6

i�1�2i � 3�

f �x� � x3 � 4x2 � 25x � 100

f �x� � x3 � 3x2 � 2x � 6

f �x� � x5 � x3 � 6xf �x� � x4 � x3 � 20x2

x

�10x3 � 51x2 � 48x � 28� � �5x � 2��3x4 � 17x3 � 10x2 � 9x � 8� � �3x � 2��2x5 � 8x3 � 4x � 1� � �x2 � 2x � 1��x4 � 2x3 � 3x2 � 8x � 4� � �x2 � 4�

f �x� � 3�x � 1�3f �x� � 2 �14 x3

f �x� � 3 � x3f �x� � �x � 2�3

y � x3

f �x� �12�x � 4�4f �x� � �2 � x 4

f �x� � x 4 � 1f �x� � �x � 3�4

y � x 4

y

an �3�1 � �0.5�n�

1 � 0.5an �

3�1 � �1.5�n�1 � 1.5

an � 3� 32�n

an � 4� 23�n

f �x� �x

�x2 � 1.

limx→c

� f �x� � g�x�� � �.c,xg�x�f �x�

gf

f �x�c,xf �x�

52. Data Analysis: Military The table shows the numbers (in thousands) of U.S. military reserve personnel for the years 1997 through 2003. (Source:U.S. Department of Defense)

A model for the data is given by

where represents the year, with correspondingto 1997.

(a) Use a graphing utility to create a scatter plot ofthe data and graph the model in the same viewingwindow. How well does the model fit the data?

(b) Use the model to predict the number of militaryreserve personnel in 2006.

(c) What is the limit of the function as approachesinfinity? Explain the meaning of the limit in thecontext of the problem. Do you think the limit isrealistic? Explain.

t

t � 7t

N �632.8 � 283.17t

1.0 � 0.27t, 7 ≤ t ≤ 13

N

Model It

Year Number, N

1997 1474

1998 1382

1999 1317

2000 1277

2001 1249

2002 1222

2003 1189

332522_1204.qxd 12/13/05 1:06 PM Page 891