L'hospital's rule exercises

1. L’Hˆ opital’s rule practice problems 21-121: Integration and Differential Equations Find the following limits. You may use L’Hˆ opital’s rule where appropriate. Be aware that L’Hˆ opital’s rule may not apply to every limit, and it may not be helpful even when it does apply. Some limits may be found by other methods. These problems are given in no particular order. (Where appropriate, sources for the problems are given in square brackets under the answer. See the end for an explanation of these references.) ex− x − 1 cosx − 1 [R 4.7 Ex.4] x3− x2− 10x − 8 5x3+ 12x2− 2x − 12 3. lim x→a [SSJ 124.29] 1 − cosθ θ2 [M 62.12] 5. lim Ans. 0. x→0xsin1 lim x→π/2 19. lim Ans. 0. 1. lim x→0 Ans. −1. x tan3x tan5x 5 3. Ans. 20. 3 5. lim x→−2 Ans. 2. [R 4.7.18] secx lnsecx x − a lnx − lna Ans. a. lim Ans. −∞. [M 62.26] 21. x→π/2+ ? ? 1 2. cscx −1 4. lim θ→0 Ans. 22. lim Ans. 0. x x→0 lim x→π/2(tanx)sin2x lim Ans. 1. [S 21.3(b)] 23. x→∞(sinhx − coshx) lim x→π/2+ tanhx tan−1x sinx sinhx ? tanx Ans. 0. x→−∞x−3ex ln(t + 2) 24. Ans. ∞. [W VIII.2.1] 6. ln(2x − π) lim x→∞ Ans. ln2. 25. 2 π. log2t 1 − lnx x/e − 1 √1 − tanx −√1 + tanx sin2x lim x→∞ Ans. 7. [R Ch.7 Rev. 115] 26. lim Ans. −1. 8. lim x→0 Ans. 1. x→e ? ex− 1−1 1 2. ex Ans. −1 [Sh 9.1.14] 9. lim x→0 Ans. 27. lim 2. x x→π [R Ch.4 rev. 116] ln(x − π/2) secx (π/2) − tan−1x x−1 √x + 10 + 3x1/3 4x2+ 3x − 1 sinx − x x3 lim x→π/2 Ans. 0. 10. lim x→∞ Ans. 1. 28. [SSJ 122 Ex.4] [W VIII.1.3 Ex.H] ex2− e4 x − 2 3√x lnx ln(tanx) sinx − cosx e−x sinx Ans. −7 Ans. −1 [SM 7.6.13] 11. lim Ans. 4e4. [R 4.7.22] lim x→−1 30. 29. x→2 30. lim 6. lim x→∞ Ans. ∞. 12. x→0 √2. tanx − x x3 1 3. lim x→π/4 Ans. [Sh 9.1.6(b)] 13. 31. lim Ans. x→0 [SM 7.6.14] lim x→∞ Ans. Does not exist. [H 4.8.18(c)] 14. x4− 4x sin(πx) 1 + tan(x/4) cos(x/2) 32(1 − ln2) π 32. lim Ans. . x→2 e−x lim x→∞ Ans. 0. 15. sinx + 2 tan−1x −π tanπ lim x→3π Ans. 1. [W VIII.1.2] 33. 1 π. 16. lim Ans. 4 4x − 1 lim Ans. e. 34. x→∞x1/(lnx) 35. lim x→1 [R 4.7.51] x→0(cosx)cscx lim Ans. 1. tan−1x cot−1x Ans. −1 [W VIII.1.7] Ans. 0. [M 62.32] lim x→−∞ 2. 17. x→π/2−cosxlntanx lim Ans. 0. [SSJ 124.17] 36. 18. lim x→0sinxlnx Ans. 1. 37. x→∞xsin(1/x)

2. h?π sin3xcot2x lncosx ?1 ? i tan−1x − π/2 + x−1 coth−1x − x−1 x1010 ex lim x→π/2 tanx Ans. 1. [SSJ 124.24] 38. 2− x lim x→∞ Ans. 1. [W VIII.4.4] 55. lim x→0+ Ans. −∞. 39. lim x→+∞ Ans. 0. [W VIII.1.5] 56. ? x2−cotx 1 3. 40. lim Ans. ?sinx ? ?1/x2 x x→0 lim x→0+ Ans. e−1/6. [R 4.7.75(a)] ?57. [S 21.1(e)] x sin−1x x2cscx ?πx 41. lim Ans. 1. ? 1 1 1 3. x→0 ?58. lim Ans. sin2x− ? Ans. −2 [R 4.7.26] x2 x→0 42. lim x→1tan lnx π. [R 4.7.75(b)] 2 √lnx?√lnx?x x lim x→+∞ Ans. ∞. [W VIII.4.11] ??59. xex− 4 + 2ex− 2x 1 + xsinπx + x/2 + 2sinπx ?√x?lnx(lnx) √x lim x→−2 43. Ans. 2e−2− 4. Ans. 0. [W VIII.3.20] x2cosx 2sin2 1 60. Compute lim 2xby two methods. √1 − x lnln(1/x) lim x→1− 44. x→0 Ans. 2. [M 62.10] x→0+(cscx)(cos−1x)/(lnx) 2sinθ − sin2θ sinθ − θcosθ ex+ x sinhx 2(ecosθ+ θ − 1) − π lnsin(−3θ) lim Ans. e−π/2. 45. 61. Finding the following limit was the first example that L’Hˆ opital gave in demon- strating the rule that bears his name. Let a be a positive constant and find √2a3x − x4− a a − 46. lim Ans. 3. θ→0 [R Ch.4 rev. 113] 3√a2x lim x→∞ Ans. 2. 47. lim x→a 4√ax3 . Ans. −2 [SSJ 124.9] lim θ→π/2 9. 48. Ans. 0. [Sh 9.1.16] xsin(1/x) lnx xcotx ex− 1 xlnx ln(1 + ax), 62. Show that L’Hˆ opital’s rule applies to the x + cosx x − cosx, but that it is of no help. Then evaluate the limit directly. lim x→0+ lim x→0+ Ans. 0. 49. limit lim x→∞ Ans. ∞. 50. Ans. 1. [R 4.7.69] lim x→0+ a > 0 Ans. −∞. [W VIII.1.10] 51. ?63. Without using L’Hˆ opital’s rule, the limit sinx x intricate geometric argument. Show that it can be evaluated easily using L’Hˆ opital’s rule. Then explain why doing so involves circular reasoning. sin−1x xcos−1x lnx cotx 2 π. 52. lim Ans. lim x→0 can be evaluated by a rather x→0 lim x→0+ lim Ans. 0. [SM 7.6.31] 53. x→∞(ax)b/(cx), a,c 6= 0 Ans. 1. 54. [R 4.7.72] References: [H] Deborah Hughes-Hallett, Andrew M. Gleason, Wil- liam G. McCallum, et al. Calculus: Single and Mul- tivariable, second edition. Wiley, 1998. [M] Ross R. Middlemiss. Differential and Integral Cal- culus, first edition. McGraw-Hill, 1940. [R] Jon Rogawski. Calculus: Early Transcendentals. Freeman, 2008. [S] Ivan S. Sokolnikoff. Advanced Calculus. McGraw- Hill, 1939. [Sh] Shahriar Shahriari. Approximately Calculus. Amer- ican Mathematical Society, 2006. [SM] Robert T. Smith and Roland B. Minton. Calculus, second edition. McGraw-Hill, 2002. [SSJ] Edward S. Smith, Meyer Salkover, and Howard K. Justice. Calculus. Wiley, 1938. [W] David V. Widder. Advanced Calculus. Hall, 1947. Prentice-