SECTION 8.6 L’Hoˆpital’s Rule 597
8.6 L’HÔPITAL’S RULE
■ Approximate limits that produce indeterminate forms. ■ Use L’Hôpital’s Rule to evaluate limits.
Indeterminate Forms and Limits In Sections 1.5 and 3.6, you studied limits such as x2 1 lim x→1 x 1 and 2x 1 lim . x→ x 1 In those sections, you discovered that direct substitution can produce an indeter- minate form such as 0 0 or . For instance, if you substitute x 1 into the first limit, you obtain x2 1 0 lim Indeterminate form x→1 x 1 0 which tells you nothing about the limit. To find the limit, you can factor and divide out like factors, as shown. x2 1 x 1 x 1 lim lim Factor. x→1 x 1 x→1 x 1 x 1 x 1 lim Divide out like factors. x→1 x 1 lim x 1 Simplify. x→1 1 1 Direct substitution 2 Simplify. For the second limit, direct substitution produces the indeterminate form , which again tells you nothing about the limit. To evaluate this limit, you can divide the numerator and denominator by x. Then you can use the fact that the limit of 1 x, as x → , is 0. 2x 1 2 1 x lim lim Divide numerator and x→ x 1 x→ 1 1 x denominator by x. 2 0 Evaluate limits. 1 0 2 Simplify. Algebraic techniques such as these tend to work well as long as the function itself is algebraic. To find the limits of other types of functions, such as exponential functions or trigonometric functions, you generally need to use a different approach. 598 CHAPTER 8 Trigonometric Functions
EXAMPLE 1 Approximating a Limit Find the limit. e3x 1 lim x→0 x
SOLUTION When evaluating a limit, you can choose from three basic approach- e3x 1 f ) x) x es. That is, you can attempt to find the limit analytically, graphically, or numer- 6 ically. For this particular limit, it is not clear how to use an analytic approach because direct substitution yields an indeterminate form. e3x 1 e3 0 1 lim lim Indeterminate form x→0 x x→0 0 0 Indeterminate form 0 2 2 0 Using a graphical approach, you can graph the function, as shown in Figure 8.36, FIGURE 8.36 and then use the zoom and trace features to estimate the limit. Using a numerical approach, you can construct a table, such as that shown below.
TRY IT 1 x 0.01 0.001 0.0001 0 0.0001 0.001 0.01
Find the limit. e3x 1 2.9554 2.9955 2.9996 ? 3.0005 3.0045 3.0455 x 1 e2x lim x→0 x From the values in the table, it appears that the limit is 3. So, from either a graph- ical or a numerical approach, you can approximate the limit to be e3x 1 lim 3. x→0 x
sin 4x f ) x) x EXAMPLE 2 Approximating a Limit 5 Find the limit. sin 4x lim x→0 x
−π π SOLUTION As in Example 1, it is not clear how to use an analytic approach because direct substitution yields an indeterminate form. 1 sin 4x sin 4 0 0 lim lim Indeterminate form FIGURE 8.37 x→0 x x→0 0 0
TRY IT 2 Using a graphical approach, you can graph the function, as shown in Figure 8.37, and then, using the zoom and trace features, you can estimate the limit to be 4. Find the limit. A numerical approach would lead to the same conclusion. So, using either a sin x 2 graphical or a numerical approach, you can approximate the limit to be lim x→0 x sin 4x lim 4. x→0 x SECTION 8.6 L’Hoˆpital’s Rule 599
L’Hôpital’s Rule
L’Hôpital’s Rule, which is named after the French mathematician Guillaume DISCOVERY Francois Antoine de L’Hôpital (1661–1704), describes an analytic approach for evaluating limits. Use a numerical or a graphical approach to approximate each limit. L’Hôpital’s Rule 22x 1 (a) lim Let a, b be an interval that contains c. Let f and g be differentiable in x→0 x a, b , except possibly at c. If the limit of f x g x as x approaches c 32x 1 (b) lim produces the indeterminate form 0 0 or , then x→0 x f x f x 42x 1 lim lim (c) lim x→c g x x→c g x x→0 x provided the limit on the right exists or is infinite. The indeterminate 52x 1 (d) lim form comes in four forms: , , , and x→0 x . L’Hôpital’s Rule can be applied to each of these forms. What pattern do you observe? Does an analytic approach have an advantage for these limits? EXAMPLE 3 Using L’Hôpital’s Rule If so, explain your reasoning. Find the limit. e3x 1 lim x→0 x
SOLUTION In Example 1, it was shown that the limit appears to be 3. Because direct substitution produces the indeterminate form 0 0, you can apply L’Hôpital’s Rule to obtain the same result. d e3x 1 e3x 1 dx lim lim Apply L’Hôpital’s Rule. x→0 x x→0 d x dx 3e3x lim Differentiate numerator and x→0 1 denominator separately. 3e3 0 Direct substitution 1 3 Simplify.
TRY IT 3 Find the limit using L’Hôpital’s Rule. 1 e2x lim x→0 x
STUDY TIP Be sure you see that L’Hôpital’s Rule involves f x g x , not the derivative of the quotient f x g x . 600 CHAPTER 8 Trigonometric Functions
STUDY TIP EXAMPLE 4 Using L’Hôpital’s Rule You cannot apply L’Hôpital’s Find the limit. Rule unless direct substitution sin 4x produces the indeterminate form lim 0 0 or . For instance,if x→0 x you were to apply L’Hôpital’s SOLUTION In Example 2, it was shown that the limit appears to be 4. Because Rule to the limit direct substitution produces the indeterminate form 0 0 x lim sin 4x 0 x→0 ex lim Indeterminate form x→0 x 0 you would incorrectly conclude you can apply L’Hôpital’s Rule to obtain the same result. that the limit is 1. Can you see why this is incorrect? d sin 4x sin 4x dx lim lim Apply L’Hôpital’s Rule. x→0 x x→0 d x TRY IT 4 dx Find the limit using L’Hôpital’s 4 cos 4x lim Differentiate numerator and Rule. x→0 1 denominator separately. sin x 2 4 cos 4 0 lim Direct substitution x→0 x 1 4 Simplify.
EXAMPLE 5 Using L’Hôpital’s Rule STUDY TIP Find the limit. Another form of L’Hôpital’s ex lim Rule states that if the limit of x→ e2x 1 f x g x as x approaches or produces the indeterminate SOLUTION Because direct substitution produces the indeterminate form form 0 0 or , then ex lim Indeterminate form → 2x f x f x x e 1 lim lim . x→ g x x→ g x you can apply L’Hôpital’s Rule, as shown. This form is used in Example 5. d ex ex dx lim lim Apply L’Hôpital’s Rule. x→ e2x 1 x→ d e2x 1 dx TRY IT 5 ex lim Differentiate numerator and Find the limit. x→ 2e2x denominator separately. 2x e 1 lim lim x Simplify. x→ e4x 1 x→ 2e 0 Evaluate limit. SECTION 8.6 L’Hoˆpital’s Rule 601
Sometimes it is necessary to apply L’Hôpital’s Rule more than once to remove an indeterminate form. This is shown in Example 6.
EXAMPLE 6 Using L’Hôpital’s Rule Repeatedly Find the limit. x2 lim x→ e x
SOLUTION Because direct substitution results in the indeterminate form x2 lim Indeterminate form x→ e x you can apply L’Hôpital’s Rule, as shown. d x2 x2 dx lim lim Apply L’Hôpital’s Rule. x→ e x x→ d e x dx 2x Differentiate numerator and lim x→ e x denominator separately. Indeterminate form After one application of L’Hôpital’s Rule, you still obtain an indeterminate form. In such cases, you can try L’Hôpital’s Rule again, as shown. d 2x 2x dx lim lim Apply L’Hôpital’s Rule. x→ e x x→ d e x dx TRY IT 6 2 Differentiate numerator and Find the limit. lim x→ e x denominator separately. x2 0 Evaluate limit. lim x→ e2x So, you can conclude that the limit is zero.
STUDY TIP Remember that even in those cases in which L’Hôpital’s Rule can be applied to determine a limit, it is still a good idea to confirm the result graphically or numerically. For instance, the table below provides a numerical confirmation of the limit in Example 6.
x 2 4 6 8 10 12
x2 0.5413 0.2931 0.0892 0.0215 0.0045 0.0008 e x 602 CHAPTER 8 Trigonometric Functions
L’Hôpital’s Rule can be used to compare the rates of growth of two functions. For instance, consider the limit in Example 5 ex lim 0. x→ e2x 1 Both of the functions f x ex and g x e2x 1 approach infinity as x → . However, because the quotient f x g x approaches 0 as x → , it follows that the denominator is growing much more rapidly than the numerator.
STUDY TIP L’Hôpital’s Rule is necessary to solve certain real-life problems such as com- pound interest problems and other business applications.
g ) x) ex f ) x) x 7 EXAMPLE 7 Comparing Rates of Growth Each of the functions below approaches infinity as x approaches infinity. Which function has the highest rate of growth? (a)f x x (b)g x ex (c) h x ln x
0 5 SOLUTION Using L’Hôpital’s Rule, you can show that each of the limits is zero. d 1 x h ) x) ln x x dx 1 lim lim lim 0 x→ ex x→ d x→ ex ex FIGURE 8.38 dx d ln x TRY IT 7 ln x dx 1 lim lim lim 0 x→ x x→ d x→ x Use a graphing utility to order x the functions according to the dx rate of growth of each function d ln x as x approaches infinity. ln x dx 1 lim lim lim 0 x→ ex x→ d x→ xex (a) f x e2x ex dx (b) g x x2 From this, you can conclude that h x ln x has the lowest rate of growth, (c) h x ln x2 f x x has a higher rate of growth, and g x ex has the highest rate of growth. This conclusion is confirmed graphically in Figure 8.38.
T AKE ANOTHER LOOK
Comparing Rates of Growth
Each of the functions below approaches infinity as x approaches infinity. In each pair, which function has the higher rate of growth? Use a graphing utility to verify your choice.
x a.f x ex, g x e x b.f x x3 1, g x x c. f x tan , g x x 2x 1 SECTION 8.6 L’Hoˆpital’s Rule 603
PREREQUISITE The following warm-up exercises involve skills that were covered in earlier sections. You will REVIEW 8.6 use these skills in the exercise set for this section.
In Exercises 1–8, find the limit, if it exists. 2x 5 1.lim x2 10x 1 2. lim x→ x→ 3 x 1 5x 2 3.lim 4. lim x→ x2 x→ x2 1 2x2 5x 7 x2 1 5.lim 6. lim x→ 3x2 12x 4 x→ x2 4 x2 5 x2 2x 1 7.lim 8. lim x→ 2x 7 x→ x 4
In Exercises 9–12, find the first derivative of the function. 9.f x cos x2 10. f x sin 5x 1 11.f x sec 4x 12. f x tan x2 2
In Exercises 13–16, find the second derivative of the function. x 13.f x sin 2x 3 14. f x cos 2 15.f x tan x 16. f x cot x
EXERCISES 8.6
In Exercises 1–6, decide whether the limit produces an indeter- sin x 9. lim minate form. x→0 5x 2x x x2 4x 3 1.lim 2. lim x 0.1 0.01 0.001 0 0.001 0.01 0.1 x→0 x x→ 7x2 2 4 sin x f x ? 3.lim 4. lim x→ x2 ex x→0 ex cos 2x 1 2xe2x ln x 10. lim 5.lim 6. lim x→0 6x x→ 3ex x→ x x 0.1 0.01 0.001 0 0.001 0.01 0.1 In Exercises 7–10, complete the table to estimate the limit numerically. f x ? e x 1 7. lim x→0 3x In Exercises 11–16, use a graphing utility to find the indicated limit graphically. x 0.1 0.01 0.001 0 0.001 0.01 0.1 ln x ex 11.lim 12. lim f x ? x→ x2 x→ x2 ln 2 x ex 1 2 13.lim 14. lim x x 12 → → 2 8. lim x 1 x 1 x 1 x 1 x→3 x2 9 x2 x 2 x2 3x 4 15.lim 16. lim x 2.9 2.99 2.999 3 3.001 3.01 3.1 x→2 x2 5x 6 x→1 x2 2x 3 f x ? 604 CHAPTER 8 Trigonometric Functions
In Exercises 17–48, use L’Hôpital’s Rule to find the limit. You may ln x 2 53. lim need to use L’Hôpital’s Rule repeatedly. x→3 x 2 e x 1 x2 x 12 17.lim 18. lim ln x 2 → → 2 54. lim x 0 x x 3 x 9 x→ 1 x 2 sin x cos 2x 1 19.lim 20. lim 2 ln x → → 55. lim x 0 5x x 0 6x x→1 ex ln x ex 21.lim 22. lim sin x → 2 → 2 56. lim x x x x x→0 x x2 x 2 x2 3x 4 23.lim 24. lim In Exercises 57–62, use L’Hôpital’s Rule to compare the rates of x→2 x2 5x 6 x→1 x2 2x 3 growth of the numerator and the denominator. 2x 1 ex 2x 1 e x 25.lim 26. lim x2 x→0 x x→0 3x 57. lim x→ e4x ln x 3x 27.lim 28. lim x3 x→ ex x→ ex 58. lim x→ e2x 4x2 2x 1 3x2 5 29.lim 30. lim ln x 4 x→ 3x2 7 x→ 2x2 11 59. lim x→ x 1 x e3x 1 31.lim 32. lim ln x 2 x→ ex x→0 2x 60. lim x→ x4 ln x ln x 1 33.lim 34. lim ln x n x→1 x2 1 x→2 x 2 61. lim x→ xm sin 2x tan 2x 35.lim 36. lim xm x→0 sin 5x x→0 x 62. lim x→ enx 2 cos x cos 2x 1 37.lim 38. lim x→ 2 3 cos x x→0 x 63. Complete the table to show that x eventually “overpowers” sin x ln x ln x 5. 39.lim 40. lim x→0 e x 1 x→1 1 x2 x 10 102 103 104 105 106 cos x 2 x 1 41.lim 42. lim 5 x→ x x→1 x 1 ln x x x x 4 2 43.lim 44. lim x→ x 1 x→0 x 64. Complete the table to show that ex eventually “overpowers” 2 2 4 x 2 x 2 x6. 45.lim 46. lim 2 x→0 x x→ ex x 1 2 48122030 e3x x2 47.lim 48. lim x→ x3 x→ ex ln x ex x6 In Exercises 49–56, find the limit. (Hint: L’Hôpital’s Rule does not apply in every case.) x2 2x 1 In Exercises 65–68, L’Hôpital’s Rule is used incorrectly. Describe 49. lim the error. x→ x2 3 e3x 1 3e3x 3x3 7x 5 65. lim lim lim 3e2x 3 50. lim x→0 ex x→0 ex x→0 x→ 4x3 2x 11 sin x 1 cos x x3 3x2 6x 8 66. lim lim 51. lim x→0 x x→0 1 x→ 1 2x3 3x2 5x 6 ex 1 ex 2x3 x2 3x 67. lim lim lim xex e 52. lim x→1 ln x x→1 1 x x→1 x→0 x4 2x3 9x2 18x x x e e 68. lim lim lim 1 1 x→ 1 e x x→ e x x→ SECTION 8.6 L’Hoˆpital’s Rule 605
In Exercises 69–72,use a graphing utility to (a) graph the function 77. Research Project Use your school’s library, the and (b) find the limit (if it exists). Internet, or some other reference source to gather data on x 2 the sales growth of two competing companies. Model the 69. lim data for both companies, and determine which company is x→2 ln 3x 5 growing at a higher rate. Write a short paper that describes 4 x your findings, including the factors that account for the 70. lim x x→ e growth of the faster-growing company. x2 4 5 71. lim x→ 2 x 2 True or False? In Exercises 78–81, determine whether the statement is true or false. If it is false, explain why or give an 3 tan x 72. lim example that shows it is false. x→0 5x x2 3x 1 2x 3 78. lim lim 3 73. Show that L’Hôpital’s Rule fails for the limit x→0 x 1 x→0 1 x lim . x 1 2 79. lim lim 1 x→ x 1 x→ 1 x x→ 1 What is the limit? 80. If 74. Use a graphing utility to graph f x lim 0, xk 1 x→ g x f x k then g x has a higher growth rate than f x . for k 1, 0.1, and 0.01. Then find f x 81. If lim 1, then g x f x . xk 1 x→ g x lim . k→0 k 75. Sales The sales (in millions of dollars per year) for the years 1994 through 2003 for two major drugstore chains are modeled by BUSINESS CAPSULE f t 1673.4 449.78t 2 28.548t 3 0.0113et Rite Aid g t 5947.9 3426.94t 841.872t 2 35.4110t 3 CVS where t is the year, with t 4 corresponding to 1994. (Source: Rite Aid Corporation; CVS Corporation) (a) Use a graphing utility to graph both models for 4 ≤ t ≤ 13.
(b) Use your graph to determine which company had the Technology,Align Courtesy of Inc. larger rate of growth of sales for 4 ≤ t ≤ 13. Two graduates of Stanford University, Zia Christi (c) Use your knowledge of functions to predict which com- and Kelsey Wirth, developed an alternative to metal pany ultimately will have the larger rate of growth. braces: a clear, removable aligner.They use 3-D Explain your reasoning. graphics to mold a person’s teeth and to prepare (d) If the models continue to be valid after 2004, when will a series of aligners that gently shift the teeth to a one company’s sales surpass the other company’s sales desired final position.The company they created, and always be greater? Align Technology, had total revenues of $6.7 million 76. Compound Interest The formula for the amount A in a in 2000 and $122.7 million in 2003, with over savings account compounded n times per year for t years at 44,000 patients using the aligners.The aligners an interest rate r and an initial deposit of P is given by provide orthodontists with an alternative treat- ment plan and provide patients with convenience. r nt A P 1 . n Use L’Hôpital’s Rule to show that the limiting formula as the number of compoundings per year becomes infinite is A Pert.