SECTION 2.5 Indeterminate Forms 95 2r2 - 9.x - 5 9. lim .12 - 25 10. lim (thy - 1 h -+0 37. If the mass of the ball is one kilogram and it is launched upward with 2x + 1 an initial velocity of 60 m/sec, then 11. my 2x2 + 3x + 1 12. lim x2 - x 1- 3 x2 _ 9 60k - 9.8 In + 1) 3x2 - 4.x - 4 H (k) = 14. lim (3 + h) - 27 3 lim 21 2 - 8 h-0 Estimate the maximum height without air resistance by investigating 42 -1 lim H (k) numerically. (h + 2)2 - 9h 15. lim 4 - 1 16. lim -4 h - 4 38. If the mass of the ball is 500 grams and it is launched upward with an Vx - 4 initial velocity of 30 m/sec, then 18. lim 21 + 4 1 16 x - 16 1- -2 12 - 312 15k - 2.45 In ( 3094 + 1) H (k) = - Al- (h + 2)2 19. Jim 20. lim yzty - 12 y-+3 y3 - 10y + 3 Estimate the maximum height without air resistance by investigating lim H(k) numerically. V2 th - 2 22. lim Vx - 4 - 2 21. lim h * -+8 x - 8 39. GU Use a plot of f (x) = - - 4 Vx - V8 -x - to estimate lim f(x) to 24. lim V5- x - 1 two decimal places. Compare with the answer obtained algebraically in THANx - V8 - x 2 - VX Exercise 23. 26. lim 40. GU Use a plot of f(x) = x _2 - x_ 4 10 estimate lim f(x) -+0+ numerically. Compare with the answer obtained algebraically in Exercise 25. col. 28. lim - cot 0 ' 140 CSC.X 0-+ 4 csc 0 41. Show numerically that for b = 3 and b = 5, lim b' - 1 - appears to 29. lim (x - 1- X2) 30. lim sin x - Cos x equal In 3 and In 5, respectively. tan x - 1 42. Show numerically that for b = 2 and b = 4, lim br - 1 appears to 22 + 2' - 20 equal In 2 and In 4, respectively. 31. lim 2' - 4 32. lim ( sec 0 - tan () 1-+ 2 In Exercises 43-48, evaluate using the identity 3. lim tan 0 - 1 tan2 0 - 1 a3 - b3 = (a - b)(a? + ab+ b? ) 2 cos x + 3 cos.x - 2 x3 - 8 43. lim x-2 x - 2 44. lim - x3 - 27 34. lim 1 -+ 3 x2 - 9 2 cos x - 1 45. lim x2 - 5x + 4 *3+ 8 35. The following limits all have the indeterminate form 0/0. One of the x3 - 1 46. lim x- -2 x2 + 6.x + 8 limits does not exist, one is equal to 0, and one is a nonzero limit. Evalu- are each limit algebraically if you can or investigate it numerically if you x* - 1 47. lim *- 1 x3 - 1 48. lim x - 27 cannot. +27 x1/3 - 3 1x2 + 3.x + 2 (a) lim - (b) lim - In Exercises 49-56, evaluate in terms of the constant a. 14-2 x + 2 x-1x-2+x-1 49. lim (2a + x) 50. lim (4ah + 7a) (c) lim 36. The following limits all have the indeterminate form co/co. One of 51. lim (41 - 2at + 3a) 52. lim (x + a)- - 4x2 x - a the limits does not exist, one is equal to 0, and one is a nonzero limit. Eval- uate each limit algebraically if you can or investigate it numerically if you 53. lim Vx - Va Va + 2h - Va cannot. x-a x-a 54 lim * -4 a) lim - (b) lim Scot.x 1-04+.x- x-+0 csc x 55. lim ( x + a)3 - a3 56. lim h a hah - a c) lim - 1+01+ - 57. Evaluate lim Vith - 1 h Hint: Set x = VI + h, express h as a function of x, and rewrite as a limit as x - 1. In Exercises 37 and 38, a ball is launched straight up in the air and is acted on by air resistance and gravity as in Example 7. The function H gives the 58. Evaluate lim With - 1 maximum height that the ball attains as a function of the air-resistance 1-0 With - 1 . Hint: Set x = 1 + h, express h as a parameter k. function of x, and rewrite as a limit as x - 1.100 CHAPTER 2 LIMITS In Exercises 29 48, evaluate the limit. 30. lim sin 4h 12. In Figure 7. is f squeezed by u and / at x = 3? At x = 2? sin 9h h-0 4h 29. lim h y = u(x) 32. lim sin h x- sin 3x v= f(x) 31. lim 4--o Sh y = 1(1) tan 4x sin 70 34. lim 1+0 9x 33) lim 8-+0 sin 30 tan 41 36. lim - 35. lim x csc 25x 1-0 f sect 140 FIGURE 7 sin 2h sin 3h 38. lim sin(z/3) 13. What does the Squeeze Theorem say about lim f(x) if the limits 37. lim :-+0 sin z lim I(x) = lim u(x) = 6 and f. u. and / are related as in Figure 8? The in- 40. lim - tan 4x equality f(x)