332522_1204.qxd 12/13/05 1:06 PM Page 883 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 ϩ 1 Exercise 52 on page 891, you f ͑x͒ ϭ . are asked to find a limit at infinity 2x to determine the number of ϭ 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 ͑x͒ ϭ Horizontal asymptote to the left x→Ϫϱ 2 1 lim f ͑x͒ ϭ Horizontal asymptote to the right x→ϱ 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 ͑x͒ ϭ L Limit as x approaches Ϫϱ x→Ϫϱ 1 and lim f ͑x͒ ϭ L Limit as x approaches ϱ x→ϱ 2 denote the limits at infinity. The first statement is read “the limit of f ͑x͒ as Ϫ ͑ ͒ x approaches ϱ is L1,” and the second is read “the limit of f x as x approaches ϱ 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 ϭ 1 ϭ 1 y1 and y2 1 Ίx Ί3 x lim ϭ 0. Limit toward the right x→ϱ 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 ϭ 0. Limit toward the left this relate to the statement at x→Ϫϱ xr the right about the infinite limit 1 lim ? Limits at infinity share many of the properties of limits listed in Section x→Ϫϱ 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 Ϫ ΃ x→ϱ x2 Algebraic Solution Graphical Solution Use the properties of limits listed in Section 12.1. Use a graphing utility to graph y ϭ 4 Ϫ 3͞x2. Then use the trace feature to determine that as x gets larger and larger,y gets 3 3 lim ΂4 Ϫ ΃ ϭ lim 4 Ϫ lim closer and closer to 4, as shown in Figure 12.31. Note that the x→ϱ x2 x→ϱ x→ϱ x2 line y ϭ 4 is a horizontal asymptote to the right. ϭ Ϫ 1 lim→ 4 3΂lim→ 2΃ y = 4 x ϱ x ϱ x 5 ϭ 4 Ϫ 3͑0͒ ϭ 4 y = 4 − 3 x2 3 − So, the limit of f ͑x͒ ϭ 4 Ϫ as x approaches ϱ is 4. 20 120 x2 −1 Now try Exercise 5. FIGURE 12.31 In Figure 12.31, it appears that the line y ϭ 4 is also a horizontal asymptote to the left. You can verify this by showing that 3 lim ΂4 Ϫ ΃ ϭ 4. x→Ϫϱ 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 ϱ for each function. complete the table below to Ϫ2x ϩ 3 Ϫ2x 2 ϩ 3 Ϫ2x3 ϩ 3 verify that a.f ͑x͒ ϭ b.f ͑x͒ ϭ c. f ͑x͒ ϭ 3x2 ϩ 1 3x2 ϩ 1 3x2 ϩ 1 1 lim ϭ 0. Solution x→ϱ 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 Ϫ2 ϩ 3 Ϫ2x ϩ 3 x x2 x a. lim ϭ lim x→ϱ 3x2 ϩ 1 x→ϱ 1 3 ϩ x 103 104 105 x2 Ϫ ϩ 1 ϭ 0 0 ϩ x 3 0 ϭ 0 Make a conjecture about 3 Ϫ2 ϩ 1 Ϫ2x2 ϩ 3 x2 lim . b. lim ϭ lim x→0 x x→ϱ 3x2 ϩ 1 x→ϱ 1 3 ϩ x2 Ϫ2 ϩ 0 ϭ 3 ϩ 0 2 ϭ Ϫ 3 3 Ϫ2x ϩ Ϫ2x3 ϩ 3 x2 c. lim ϭ lim x→ϱ 3x2 ϩ 1 x→ϱ 1 3 ϩ 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. 5x͑x Ϫ 3͒ a. lim Now try Exercise 13. x→ϱ 2x 3 4x Ϫ 5x In Example 2, observe that when the degree of the numerator is less than the b. lim 4 ϩ 2 Ϫ x→ϱ 8x 3x 2 degree of the denominator, as in part (a), the limit is 0. When the degrees of the Ϫ 2 ϩ 6x 1 numerator and denominator are equal, as in part (b), the limit is the ratio of the c. lim 2 ϩ Ϫ x→ϱ 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 f͑x͒ ϭ N͑x͒͞D͑x͒, where ͑ ͒ ϭ n ϩ . ϩ ͑ ͒ ϭ m ϩ . ϩ N x anx a0 and D x bm x b0. The limit of f͑x͒ as x approaches positive or negative infinity is as follows. 0, n < m lim f ͑x͒ ϭ an x→±ϱ Ά , n ϭ 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 ϭ 0.50x ϩ 5000. The average cost C per card is given by C 0.50x ϩ 5000 C ϭ ϭ . x x Find the average cost per card when (a)x ϭ 1000, (b)x ϭ 10,000, and (c)x ϭ 100,000. (d) What is the limit of C as x approaches infinity? Solution a. When x ϭ 1000, the average cost per card is 0.50͑1000͒ ϩ 5000 C ϭ x ϭ 1000 1000 ϭ $5.50. b. When x ϭ 10,000, the average cost per card is Average Cost 0.50͑10,000͒ ϩ 5000 C C ϭ x ϭ 10,000 10,000 6 ϭ $1.00. 5 c. When x ϭ 100,000, the average cost per card is 4 0.50͑100,000͒ ϩ 5000 C 0.50x + 5000 ϭ ϭ 3 C = = C x 100,000 x x 100,000 (in dollars) 2 ϭ 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 ϩ 5000 lim ϭ $0.50. x → ϱ Number of cards x→ϱ x →ϱ As x , the average cost per card The graph of C is shown in Figure 12.32.