EXERCISES 4.5 Finding Limits in Two Ways In Exercises 1-6, use l'Hopital's Rule to evaluate the limit. Then cos 0 - 1 evaluate the limit using a method studied in Chapter 2. 43. lim e" - (1 + h ) 0-20 20 - 0- 1 44. lim 1. lim *+ 2 2. lim Sin 5x 45. lim + 12 x--2x2 - 4 X 1 00 et - 1 46. lim x2ex 5x2 -- 3x 3. lim - 4. lim- x3 - 47. lim * - sin x (ex - 1) 8-+00 7x2 + 1 x-14x3 - x - x-+0 * tan x 48. 4 0 x sin x - Cos x . lim - 2x2 + 3x 49. lim 0 - sin 0 cos 0 5. lim 0 - 0 tan 0 - 0 50. lim Sin 3x - 3x + x2 x-10ox + x+ 1 sin x sin 2x Applying I'Hopital's Rule Indeterminate Powers and Products Use I'Hopital's rule to find the limits in Exercises 7-50. Find the limits in Exercises 51-66 51. lim x1/(1-x) x- 1+ 52. lim x1/(x-1) 7. lim 2 x2 - 25 8. lim x - 1+ x-+2x2 - 4 * - - 5 x+ 5 53. lim (In x) 1/x 54. lim (In x) 1/(-e) 9. lim - 1 - 41 + 15 10. lim - 3+3+ 3 55. lim x 1/In x 1- - 312 - 1-12 1- - 14+3 - + + 3 56. lim x1/in z 5x3 - 2x 12. lim - x - 8x2 57. lim (1 + 2x) 1/(2 In x) 11. lim = 58. lim (ex + x)1/x *-+00 7x3 + 3 x-+00 12x2 + 5x x - 59. lim x 13. lim- sin 12 14. lim Sin St 60. lim (1 + 2 )" t x -0+ 1- 0 2t 15. lim- 16. lim Sin x - x 61. lim (* + 2 ) 62. lim (x2 + 1 )1/2 x-0COS x - 1 x-0 *-+00 20 - TV 30 + TV 63. lim x2 In x 64. lim x (In x)2 17. lim 18. 9-+17/2 COS (2TT - 0) lim 0-TT/3 sin (0 + (T / 3)) 19. lim - sin 0 65. lim x tan x -+ 0+ " -x) 66. lim sin x . In x 20. lim x-0+ 0-#/21 + cos 20 x-+i In x - sin Trx Theory and Applications 21. lim In (csc x) z-0In (sec x) 22. 1 *-NTT / 2 ( x - ( 15 / 2) )2 L'Hopital's Rule does not help with the limits in Exercises 67-74. Try it-you just keep on cycling. Find the limits some other way. 23. lim- I(1 - cos z) 24. lim - t sin t 1+0 t - sin 1 1 01 - cos t 67. lim - Vox + 1 68. lim Vx *-+0o Vx + 1 .x-0* V sinx 25. lim sec x 26. - x) tan x 69. lim sec x X-(7/2) ( TT / 2 ) - x-(17/2) - tan x 70. lim cot x x- 0+ CSC X 27. lim 3sin 0 - 1 (1/2)0 - 1 2x - 3x 0 -+ 0 28. lim 0-0 71. lim 2X 2* + 4x * -co 3* + 4x 72. lim *- - 005x - 2x 29. lim X2" er -20 2x - 1 3* - 1 30. lim x -+ 02* - 1 73. lim 74. lim- * -too xet *- Ot e-1/2 31. lim- In (x + 1) log2 x 75. Which one is correct, and which one is wrong? Give reasons for 10 log2x 32. lim *-+0 10g3 (x + 3) your answers . 33. lim n (x2 + 2x) In (ex - 1) x - 3 34. lim a. lim = lim 2x 6 b. lim * - 3 In x x -+ 0 In x * + 3 x2 - 3 * - 3 x 2 - 3 0 =0 35. lim VSy + 25 -5 Vay + a - a 76. Which one is correct, and which one is wrong? Give reasons for 36. lim a >0 1- your answers . 2x - 2 -+00 37. lim (In 2x - In (x + 1)) 38. lim (In x - In sin x) a. lim * - 2x = lim x-+0x- sin x x-02x - Cos x 39. lim (In x )2 40. lim 3 x + 1 1 2 = lim ? 2 1-0* In (sin x) sin x x-02 + sin x 2 +0 41. lim 42. lim (csc x - cot x + cos x) b. lim x2 - 2x - lim ? 2x - 2 - 2 0 - 1 - = 2 In x x-+02x - Cos x * - 0+ x-0x2 - sin x