5128_Ch08_434-471

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Section 8.2 L’Hôpital’s Rule 449 We apply l’Hôpital’s Rule to the previous expression. lim ln f x lim x→ x→ lim x→ ln ( 1 1 x ) 1 x ( 1 x 2) 1 1 1 x x 12 0 0 Differentiate numerator and denominator. Therefore, 1 lim 1 x→ 1 1 x x lim x→ ( 1 1 x ) lim f x lim e ln f x e 1 e. x→ x→ Now try Exercise 21. EXAMPLE 9 Working with Indeterminate Form 0 0 Determine whether lim x→0 x x exists and find its value if it does. [–3, 3] by [–1, 3] Figure 8.11 The graph of y x x . (Example 9) SOLUTION Investigate Graphically Figure 8.11 suggests that the limit exists and has a value near 1. Solve Analytically The limit leads to the indeterminate form 0 0 . To convert the problem to one involving 00, we let f x x x and take the logarithm of both sides. ln f x x ln x l n x 1x Applying l’Hôpital’s Rule to ln x1x we obtain lim ln f x lim l n x x→0 x→0 1x lim 1x x→0 1x 2 lim x 0. x→0 Therefore, Differentiate. lim x x lim f x lim x→0 x→0 x→0 eln f x e 0 1. Now try Exercise 23. EXAMPLE 10 Working with Indeterminate Form 0 Find lim x→ x 1x . SOLUTION Let f x x 1x . Then ln f x ln x . x continued
450 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals Applying l’Hôpital’s Rule to ln f x we obtain lim ln f x lim ln x x→ x→ x lim 1 x x→ 1 Differentiate. Therefore, lim x→ 1 x 0. lim x→ x1x lim f x lim e ln f x e 0 1. x→ x→ Now try Exercise 25. Quick Review 8.2 (For help, go to Sections 2.1 and 2.2.) In Exercises 1 and 2, use tables to estimate the value of the limit. 1. lim x→ ( 1 0 .1 1.1052 2. lim x )x x1ln x 2.7183 x→0 In Exercises 3–8, use graphs or tables to estimate the value of the limit. 3. lim x→0 ( x 1 1 x ) x 1 4. lim x→1 ( 1 1 x ) t 1 5. lim 2 6. lim 4 x 2 1 2 t→1 t 1 x→ x 1 7. lim sin 3x 3 8. lim x→0 x tan u 1 u→p2 2 tan u In Exercises 9 and 10, substitute x 1h to express y as a function of h. 9. y x sin 1 x x y sin h y (1 h) 1h h 10. y ( 1 1 x ) Section 8.2 Exercises In Exercises 1–4, estimate the limit graphically and then use l’Hôpital’s Rule to find the limit. 2 1. lim x→2 x x2 2. lim sin (5x) 4 x→0 x 3. lim 2 x 2 4. lim 3 x 1 x→2 x 2 x→1 x 1 In Exercises 5–8, apply the stronger form of l’Hôpital’s Rule to find the limit. 5. lim 1 cos x 1 sin 12 6. lim 14 x→0 x2 u→p/2 1 cos (2) cos t 1 x 7. lim t→0 e t –1 8. lim 2 4x 4 t 1 x→2 x 3 16 12x 16 In Exercises 9–12, use l’Hôpital’s Rule to evaluate the one-sided limits. Support your answer graphically. 9. (a) lim s in 4x 2 (b) lim x→0 sin 2x s in 4x 2 x→0 sin 2x 10. (a) lim tan x 1 (b) lim x→0 x tan x 1 x→0 x 11. (a) lim si n x (b) lim x→0 x3 si n x x→0 x3 12. (a) lim ta n x (b) lim x→0 x2 ta n x x→0 x2 In Exercises 13–16, identify the indeterminate form and evaluate the limit using l’Hôpital’s Rule. Support your answer graphically. csc x Left ()(), 13. lim 14. lim x→p 1 cot x 1 sec x Left ()(), right ()(), x→p2 tan x right ()(), limit 1 limit 1 15. lim ln ( x 1) ()(), limit ln 2 x→ log2 x 16. lim 5 x2 3x x→ 7x2 ()(), limit 57 1 In Exercises 17–26, identify the indeterminate form and evaluate the limit using l’Hôpital’s Rule. 17. lim (x ln x) • 0, 0 18. lim x→0 x→ x tan 1 x • 0, 1 19. lim csc x cot x cos x 20. lim ln (2x) ln x 1 x→0 x→ ,1 21. lim e x x 1x 22. lim x 1x1 , ln 2 x→0 1 , e 2 x→1 1 , e 23. lim x→1 x 2 2x 1 x1 0 0 ,1 24. lim x→0 sin x x 0 0 ,1 25. lim x→0 ( x 1 1 x ) 0 ,1 26. lim x→ ln x 1x 0 ,1 In Exercises 27 and 28, (a) complete the table and estimate the limit. (b) Use l’Hôpital’s Rule to confirm your estimate. x 5 27. lim f x, f x ln x→ x x | 10 | 10 2 | 10 3 | 10 4 | 10 5 f x | | | | | 28. lim f x, f x x sin x x→0 x3 x | 10 0 | 10 1 | 10 2 | 10 3 | 10 4 f x | | | | |
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5128_Ch08_434-471

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