19. 0' that approaches (a) 1, (b) 0, (c) co. 20. 1" that approaches (a) 1, (b) e, (c) co. Skills Find the roots, discontinuities, and horizontal and vertical 51. lim (x-1/3 - x-1/2) asymptotes of the functions in Exercises 23-34. Support your x -+ 0+ 52. lim (x-1/3 - x-1/2) *-+00 answers by explicitly computing any relevant limits. 53. lim 54. lim x2 - 2x - 3 2x2 - 1 1-200 x2 - x-1 x-+0+ x2 - x-1 23. f ( x ) = . x - 3 24. f ( x ) = - x2 - 2x +1 x/12 _ x8/3 55. x 7/2 _ x8/3 lim 56. lim x-+ 0+ x2 x2 25. f ( x ) = (x+1)(x-2) ( x+1) (x-2)2 X-+ 0 (x - 2) ( x+ 2 ) 26. f(x) = _ (x-2) (x+2) 4x - 3x 57. lim 2x - 4-* (x+1)(x-2) (x+1) (x-2) 5x 58. lim 27. f (x) = 28. f(x) = X-+0 3x (x- 2)2 ( x+ 2 ) ( x - 2) (x + 2 ) 2 3x - 5x 2 59. lim 4(3) 60. 4x lim 29. f(x) = x - -00 X-00 2 + 3* 4 + e - 2x 30. f(x) = 2 + 3x 2p1.5x 2x -4* 4x - 6(2*) +5 61. 1 - 5e2x lim 62. x-+00 3e2x + 1.5x lim 31. f(x) = 3x 32. f ( x ) = x-+00 3ex + 4exx 1 - 2x 63. lim In(x2 - 9) 33. f(x) = tan (3x) + 1 64. lim In 34. f ( x) = - x -+0+ tan-1 x 65. lim ( Inx2 - In(2x + 1)) 66. lim ( In 3x - In 2x) Calculate each limit in Exercises 35-80. 35. lim -4x-3 36. lim 2x-3/4 1 - cos 2x 57. sin 3x X-+ 0 x-+ 0 lim 68. x -+ 0 7x lim X-0 5x 37. lim 2x 4/3 38. lim -5x3/5 X X-+ 00 x - - DO 69. lim -01 - cos x 70. 3 sin x + x lim 39. lim (Vx - x) 40. lim (x4 - x5) 71. lim sin 3x 72. lim sin (3x2) x-+0x3 - x * -+ 20 41. lim (-3x5 + 4x + 11) 42. lim (5 -2x + 3x3) x-0 x3 - x 13. lim x2 + 8x + 16 x + 1 lim x2 csc 3x 74. lim x2 cotx 44. * -+ -4 (x + 4) 2 (x + 1 ) lim 73. x-2 (x - 2) 2 x-0+ 1 - cos 2x x-0 sinx 45. lim *2 + 1 x +4 75. sec x tan x 46. lim lim X -+ 76. lim 3x2 cot2 x x-0 x(x - 1) x- -4x2 + 8x 4 16 17. lim - x - 1 77. lim(1 + x)2/x x-+ 0 x2 - x 48. lim- x -+0' 78. lim(1 + 2x) 3/x x-+1 x2 - 2x + 1 19. lim (3x + 1)2(x - 1) 1 -272 79. lim (1+ 2 ) 1 - x3 50. lim *--00 (3 - x) (3+ 4x) 80. lim (1