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Section 12.4 Limits at Infinity and Limits of Sequences 883
12.4 Limits at Infinity and Limits of Sequences
What you should learn •Evaluate limits of functions Limits at Infinity and Horizontal Asymptotes at infinity. As pointed out at the beginning of this chapter, there are two basic problems in •Find limits of sequences. calculus: finding tangent lines and finding the area of a region. In Section 12.3, you should learn it you saw how limits can be used to solve the tangent line problem. In this section Why and the next, you will see how a different type of limit, a limit at infinity, can be Finding limits at infinity is used to solve the area problem. To get an idea of what is meant by a limit at useful in many types of real-life infinity, consider the function given by applications. For instance, in x 1 Exercise 52 on page 891, you f x . are asked to find a limit at infinity 2x to determine the number of 1 military reserve personnel in the The graph of f is shown in Figure 12.30. From earlier work, you know that y 2 future. is a horizontal asymptote of the graph of this function. Using limit notation, this can be written as follows. 1 lim f x Horizontal asymptote to the left x→ 2 1 lim f x Horizontal asymptote to the right x→ 2 1 These limits mean that the value of f x gets arbitrarily close to 2 as x decreases or increases without bound. y
3 x + 1 © Karen Kasmauski/Corbis f(x) = 2 2x y = 1 2 1
x −3 −2 −1 123
−2
−3
FIGURE 12.30
Definition of Limits at Infinity
Iff is a function and L1 and L2 are real numbers, the statements lim f x L Limit as x approaches x→ 1 and lim f x L Limit as x approaches x→ 2 denote the limits at infinity. The first statement is read “the limit of f x as x approaches is L1,” and the second is read “the limit of f x as x approaches is L2.” 332522_1204.qxd 12/13/05 1:06 PM Page 884
884 Chapter 12 Limits and an Introduction to Calculus Exploration To help evaluate limits at infinity, you can use the following definition. Use a graphing utility to graph Limits at Infinity the two functions given by If r is a positive real number, then 1 1 y1 and y2 1 x 3 x lim 0. Limit toward the right x→ xr in the same viewing window. Furthermore, if xr is defined when x < 0, then Why doesn’t y1 appear to the left of the y -axis? How does 1 lim 0. Limit toward the left this relate to the statement at x→ xr the right about the infinite limit 1 lim ? Limits at infinity share many of the properties of limits listed in Section x→ xr 12.1. Some of these properties are demonstrated in the next example.
Example 1 Evaluating a Limit at Infinity
Find the limit. 3 lim 4 x→ x2 Algebraic Solution Graphical Solution Use the properties of limits listed in Section 12.1. Use a graphing utility to graph y 4 3 x2. Then use the trace feature to determine that as x gets larger and larger,y gets 3 3 lim 4 lim 4 lim closer and closer to 4, as shown in Figure 12.31. Note that the x→ x2 x→ x→ x2 line y 4 is a horizontal asymptote to the right. 1 lim→ 4 3 lim→ 2 y = 4 x x x 5 4 3 0 4 y = 4 − 3 x2 3 − So, the limit of f x 4 as x approaches is 4. 20 120 x2 −1 Now try Exercise 5. FIGURE 12.31
In Figure 12.31, it appears that the line y 4 is also a horizontal asymptote to the left. You can verify this by showing that 3 lim 4 4. x→ x2 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 the top of the page. 332522_1204.qxd 12/13/05 1:06 PM Page 885
Section 12.4 Limits at Infinity and Limits of Sequences 885 Exploration Example 2 Comparing Limits at Infinity Use a graphing utility to Find the limit as x approaches for each function. complete the table below to 2x 3 2x 2 3 2x3 3 verify that a.f x b.f x c. f x 3x2 1 3x2 1 3x2 1 1 lim 0. Solution x→ x In each case, begin by dividing both the numerator and denominator by x2, the x 100 101 102 highest-powered term in the denominator. 1 2 3 2x 3 x x2 x a. lim lim x→ 3x2 1 x→ 1 3 x 103 104 105 x2 1 0 0 x 3 0 0 Make a conjecture about 3 2 1 2x2 3 x2 lim . b. lim lim x→0 x x→ 3x2 1 x→ 1 3 x2 2 0 3 0 2 3 3 2x 2x3 3 x2 c. lim lim x→ 3x2 1 x→ 1 3 x2
Have students use these observations In this case, you can conclude that the limit does not exist because the from Example 2 to predict the following numerator decreases without bound as the denominator approaches 3. limits. 5x x 3 a. lim Now try Exercise 13. x→ 2x 3 4x 5x In Example 2, observe that when the degree of the numerator is less than the b. lim 4 2 x→ 8x 3x 2 degree of the denominator, as in part (a), the limit is 0. When the degrees of the 2 6x 1 numerator and denominator are equal, as in part (b), the limit is the ratio of the c. lim 2 x→ 3x x 2 coefficients of the highest-powered terms. When the degree of the numerator is Then ask several students to verify the greater than the degree of the denominator, as in part (c), the limit does not exist. predictions algebraically, several other This result seems reasonable when you realize that for large values of x, the students to verify the predictions numerically, and several more students highest-powered term of a polynomial is the most “influential” term. That is, a to verify the predictions graphically. polynomial tends to behave as its highest-powered term behaves as x approaches Lead a discussion comparing the results. positive or negative infinity. 332522_1204.qxd 12/13/05 1:06 PM Page 886
886 Chapter 12 Limits and an Introduction to Calculus
Limits at Infinity for Rational Functions Consider the rational function f x N x D x , where n . . . m . . . N x anx a0 and D x bm x b0. The limit of f x as x approaches positive or negative infinity is as follows.
0, n < m
lim f x an x→± , n m bm
If n > m, the limit does not exist.
Example 3 Finding the Average Cost
Consider asking your students to You are manufacturing greeting cards that cost $0.50 per card to produce. Your identify the practical interpretation initial investment is $5000, which implies that the total cost C of producing x of the limit in part (d) of Example 3. cards is given by C 0.50x 5000. The average cost C per card is given by C 0.50x 5000 C . x x Find the average cost per card when (a)x 1000, (b)x 10,000, and (c)x 100,000. (d) What is the limit of C as x approaches infinity? Solution a. When x 1000, the average cost per card is 0.50 1000 5000 C x 1000 1000 $5.50. b. When x 10,000, the average cost per card is Average Cost 0.50 10,000 5000 C C x 10,000 10,000 6 $1.00. 5 c. When x 100,000, the average cost per card is 4 0.50 100,000 5000 C 0.50x + 5000 3 C = = C x 100,000 x x 100,000 (in dollars) 2
erage cost per card $0.55.
Av 1 d. As x approaches infinity, the limit of C is x y = 0.5 20,000 60,000 100,000 0.50x 5000 lim $0.50. x → Number of cards x→ x → As x , the average cost per card The graph of C is shown in Figure 12.32. approaches $0.50. FIGURE 12.32 Now try Exercise 49. 332522_1204.qxd 12/13/05 1:06 PM Page 887
Section 12.4 Limits at Infinity and Limits of Sequences 887
Limits of Sequences Limits of sequences have many of the same properties as limits of functions. For n instance, consider the sequence whose n th term is an 1 2 . 1 1 1 1 1 , , , , , . . . 2 4 8 16 32 As n increases without bound, the terms of this sequence get closer and Technology closer to 0, and the sequence is said to converge to 0. Using limit notation, you There are a number of ways to use can write a graphing utility to generate the 1 terms of a sequence. For instance, lim 0. n→ n you can display the first 10 terms 2 of the sequence The following relationship shows how limits of functions of x can be used to 1 evaluate the limit of a sequence. a n 2n using the sequence feature or Limit of a Sequence the table feature. Letf be a function of a real variable such that lim f x L. x→ If an is a sequence such that f n an for every positive integer n, then lim a L. n→ n
Another sequence that diverges is A sequence that does not converge is said to diverge. For instance, the terms 1 of the sequence 1, 1, 1, 1, 1, . . . oscillate between 1 and 1. Therefore, the an . You might want your n 1 4 sequence diverges because it does not approach a unique number. students to discuss why this is true.
Example 4 Finding the Limit of a Sequence
Find the limit of each sequence. (Assume n begins with 1.) 2n 1 a. a n n 4 2n 1 b. b n n2 4 2n2 1 c. c n 4n2 Solution
2n 1 3 5 7 9 11 13 a. lim 2 , , , , , , . . . → 2 n→ n 4 5 6 7 8 9 10 You can use the definition of 2n 1 3 5 7 9 11 13 b. lim 0 , , , , , , . . . → 0 limits at infinity for rational n→ n2 4 5 8 13 20 29 40 functions on page 886 to verify 2n2 1 1 3 9 19 33 51 73 1 the limits of the sequences in c. lim , , , , , , . . . → n→ 4n2 2 4 16 36 64 100 144 2 Example 4. Now try Exercise 33. 332522_1204.qxd 12/13/05 1:06 PM Page 888
888 Chapter 12 Limits and an Introduction to Calculus
In the next section, you will encounter limits of sequences such as that shown in Example 5. A strategy for evaluating such limits is to begin by writing the n th term 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.
Example 5 Finding the Limit of a Sequence
Find the limit of the sequence whose n th term is 8 n n 1 2n 1 a . n n3 6 Algebraic Solution Numerical Solution
Begin by writing the n th term in standard rational function Construct a table that shows the value of an as n form—as the ratio of two polynomials. becomes larger and larger, as shown below. 8 n n 1 2n 1 an Write original n th term. n3 6 n an 8 n n 1 2n 1 18 Multiply fractions. 6n3 10 3.08 8n3 12n2 4n 100 2.707 Write in standard rational form. 3n3 1000 2.671 From this form, you can see that the degree of the numerator is 10,000 2.667 equal to the degree of the denominator. So, the limit of the sequence is the ratio of the coefficients of the highest-powered terms. From the table, you can estimate that as n approaches , a gets closer and closer to 2.667 8. 8n3 12n2 4n 8 n 3 lim n→ 3n3 3 Now try Exercise 43.
W RITING ABOUT MATHEMATICS Comparing Rates of Convergence In the table in Example 5 above, the value of 8 an approaches its limit of 3 rather slowly. (The first term to be accurate to three decimal places is a4801 2.667. ) Each of the following sequences converges to 0. Which converges the quickest? Which converges the slowest? Why? Write a short paragraph discussing your conclusions.
1 1 1 a.a b.b c. c n n n n2 n 2n 1 2n d.d e. h n n! n n! 332522_1204.qxd 12/13/05 1:06 PM Page 889
Section 12.4 Limits at Infinity and Limits of Sequences 889
12.4 Exercises
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.
In Exercises 1–4, match the function with its graph, using t 2 4y4 13.lim 14. lim horizontal asymptotes as aids. [The graphs are labeled (a), t→ t 3 y→ y 2 3 (b), (c), and (d).] 1 2t 6t2 15. lim (a)y (b) y t→ 5 3t 4t2 3 6 2x2 5x 12 16. lim 2 2 4 x→ 1 6x 8x 2 x2 3 2x2 6 17.lim 18. lim x x x→ 2 x 2 x→ x 1 2 −2 −1 123 −4 −2 246 −1 x 2x2 19.lim 4 20. lim 7 −2 −4 x→ x 1 2 x→ x 3 2 −3 −6 1 5t 21. lim t→ 3t2 t 2 (c)y (d) y x 3x2 6 6 22. lim x→ 2x 1 x 3 2
In Exercises 23–28, use a graphing utility to graph the x x function and verify that the horizontal asymptote −4 −2 246 −4 −2 246 −2 −2 corresponds to the limit at infinity. −4 −4 3x x2 −6 −6 23.y 24. y 1 x x2 4 2 2 2x 2x 1 4x x 25.y 26. y 1.f x 2. f x 2 2 x2 1 x2 1 1 x x 1 1 1 3 1 3.f x 4 4. f x x 27.y 1 28. y 2 x2 x x2 x
In Exercises 5–22, find the limit (if it exists). If the limit does Numerical and Graphical Analysis In Exercises 29–32, not exist, explain why. Use a graphing utility to verify your (a) complete the table and numerically estimate the limit result graphically. as x approaches infinity, and (b) use a graphing utility to graph the function and estimate the limit graphically. 3 5 5.lim 6. lim x→ 2 → x x 2x x 100 101 102 103 104 105 106 3 x 1 6x 7.lim 8. lim f x x→ 3 x x→ 1 5x 4x 3 1 2x 9.lim 10. lim 29. f x x x2 2 x→ 2x 1 x→ x 2 30. f x 3x 9x2 1 3x2 4 3x2 1 11.lim 12. lim 2 x→ 1 x2 x→ 4x2 5 31. f x 3 2x 4x x 32. f x 4 4x 16x2 x 332522_1204.qxd 12/13/05 1:06 PM Page 890
890 Chapter 12 Limits and an Introduction to Calculus
In Exercises 33–42, write the first five terms of the sequence (a) What is the limit of this function as t approaches and find the limit of the sequence (if it exists). If the limit infinity? does not exist, explain why. Assume n begins with 1. (b) Use a graphing utility to graph the function and verify n 1 n the result of part (a). 33.a 34. a n n2 1 n n2 1 (c) Explain the meaning of the limit in the context of the problem. n 4n 1 35.a 36. a n 2n 1 n n 3 49. Average Cost The cost function for a certain model of personal digital assistant (PDA) is given by n2 4n2 1 37.a 38. a C 13.50x 45,750, where C is measured in dollars and n n 3n 2 2n x is the number of PDAs produced. n 1 ! 3n 1 ! 39.a 40. a (a) Write a model for the average cost per unit produced. n n! n 3n 1 ! (b) Find the average costs per unit when x 100 and n n 1 1 1 x 1000. 41.an 42. an 2 n n (c) Determine the limit of the average cost function as x approaches infinity. Explain the meaning of the limit in In Exercises 43–46, find the limit of the sequence. Then the context of the problem. verify the limit numerically by using a graphing utility to 50. Average Cost The cost function for a company to recycle complete the table. x tons of material is given by C 1.25x 10,500, where C is measured in dollars. 0 1 2 3 4 5 6 n 10 10 10 10 10 10 10 (a) Write a model for the average cost per ton of material
an recycled. (b) Find the average costs of recycling 100 tons of material 1 1 n n 1 and 1000 tons of material. 43. a n n n n 2 (c) Determine the limit of the average cost function as x approaches infinity. Explain the meaning of the limit in 4 4 n n 1 44. a n the context of the problem. n n n 2 51. Data Analysis: Social Security The table shows the 16 n n 1 2n 1 average monthly Social Security benefits B (in dollars) for 45. an 3 n 6 retired workers aged 62 or over from 1997 to 2003. n n 1 1 n n 1 2 (Source: U.S. Social Security Administration) 46. a n n2 n4 2
47. Oxygen Level Suppose that f t measures the level Year Benefit, B of oxygen in a pond, where f t 1 is the normal (unpolluted) level and the time t is measured in weeks. 1997 765 When t 0, organic waste is dumped into the pond, and as 1998 780 the waste material oxidizes, the level of oxygen in the pond 1999 804 is given by 2000 844 t 2 t 1 2001 874 f t . t 2 1 2002 895 (a) What is the limit of this function as t approaches 2003 922 infinity? (b) Use a graphing utility to graph the function and verify A model for the data is given by the result of part (a). 199.0 999.46t B , 7 ≤ t ≤ 13 (c) Explain the meaning of the limit in the context of the 1.0 1.43t 0.032t 2 problem. where t represents the year, with t 7 corresponding to 48. Typing Speed The average typing speed S (in words per 1997. minute) for a student after t weeks of lessons is given by (a) Use a graphing utility to create a scatter plot of the data 100t 2 and graph the model in the same viewing window. How S , t > 0. 65 t 2 well does the model fit the data? 332522_1204.qxd 12/13/05 1:06 PM Page 891
Section 12.4 Limits at Infinity and Limits of Sequences 891
(b) Use the model to predict the average monthly benefit 57. Think About It Find the functions f and g such that both in 2006. f x and g x increase without bound as x approaches c, but lim f x g x . (c) Discuss why this model should not be used to predict the x→c average monthly Social Security benefits in future years. 58. Think About It Use a graphing utility to graph the function given by x Model It f x . x2 1 52. Data Analysis: Military The table shows the How many horizontal asymptotes does the function appear numbers N (in thousands) of U.S. military reserve to have? What are the horizontal asymptotes? personnel for the years 1997 through 2003. (Source: U.S. Department of Defense) Exploration In Exercises 59–62, create a scatter plot of the terms of the sequence. Determine whether the sequence Year Number, N converges or diverges. If it converges, estimate its limit. n n 59.a 4 2 60. a 3 3 1997 1474 n 3 n 2 3 1 1.5 n 3 1 0.5 n 1998 1382 61.a 62. a n 1 1.5 n 1 0.5 1999 1317 2000 1277 Skills Review 2001 1249 2002 1222 In Exercises 63 and 64, sketch the graphs of y and each 2003 1189 transformation on the same rectangular coordinate system. 63. y x 4 A model for the data is given by (a)f x x 3 4 (b) f x x 4 1 632.8 283.17t 1 N , 7 ≤ t ≤ 13 (c)f x 2 x 4 (d) f x x 4 4 1.0 0.27t 2 64. y x3 where t represents the year, with t 7 corresponding (a)f x x 2 3 (b) f x 3 x3 to 1997. (c)f x 2 1 x3 (d) f x 3 x 1 3 (a) Use a graphing utility to create a scatter plot of 4 the data and graph the model in the same viewing In Exercises 65–68, divide using long division. window. How well does the model fit the data? (b) Use the model to predict the number of military 65. x4 2x3 3x2 8x 4 x2 4 reserve personnel in 2006. 66. 2x5 8x3 4x 1 x2 2x 1 (c) What is the limit of the function as t approaches 67. 3x4 17x3 10x2 9x 8 3x 2 infinity? Explain the meaning of the limit in the 68. 10x3 51x2 48x 28 5x 2 context of the problem. Do you think the limit is realistic? Explain. In Exercises 69–72, find all the real zeros of the polynomial function. Use a graphing utility to graph the function and verify that the real zeros are the x -intercepts of the graph Synthesis of the function.
4 3 2 5 3 True or False? In Exercises 53–56, determine whether the 69.f x x x 20x 70. f x x x 6x statement is true or false. Justify your answer. 71. f x x3 3x2 2x 6 3 2 53. Every rational function has a horizontal asymptote. 72. f x x 4x 25x 100 54. If f x increases without bound as x approaches c, then the In Exercises 73–76, find the sum. limit of f x exists. 6 4 55. If a sequence converges, then it has a limit. 73. 2i 3 74. 5i2 56. When the degrees of the numerator and denominator of a i 1 i 0 rational function are equal, the limit does not exist. 10 8 3 75.15 76. 2 k 1 k 0 k 1