Question

Use the properties of limits to help decide whether each limi exists. If a limit exists, find its value. 31. lim x→3 ​ x−3 x 2 −9 ​ 32. lim x→−2 ​ x+2 x 2 −4 ​ 33. lim x→1 ​ x 2 −1 5x 2 −7x+2 ​ 34. lim x→−3 ​ x 2 +x−6 x 2 −9 ​ 35. lim x→−2 ​ x+2 x 2 −x−6 ​ 36. lim x→5 ​ x−5 x 2 −3x−10 ​ 37. lim x→0 ​ x 1/(x+3)−1/3 ​ 38. lim x→0 ​ x −1/(x+2)+1/2 ​ 39. lim x→25 ​ x−25 x ​ −5 ​ 40. lim x→36 ​ x−36 x ​ −6 ​ 41. lim h→0 ​ h (x+h) 2 −x 2 ​ 42. lim h→0 ​ h (x+h) 3 −x 3 ​ 43. lim x→−1 ​ 7x−1 3x ​ 44. lim x→−[infinity] ​ 4x−5 8x+2 ​ 46. lim x→[infinity] ​ 3x 2 +2 x 2 +2x−5 ​ Let f(x)= ⎩ ⎨ ⎧ ​ x−1 2 x+3 ​ if x<3 if 3≤x≤5 if x>5 ​ (a) Find lim x→3 ​ f(x) (b) Find lim x→5 ​ f(x).

Answer 1

To determine whether each limit exists, we can evaluate the function as x approaches the given values.

31. lim x→3 (x-3)/(x^2-9) = 0/0 (indeterminate form)

32. lim x→-2 (x+2)/(x^2-4) = 0/0 (indeterminate form)

33. lim x→1 (x^2-1)/(5x^2-7x+2) = 0/0 (indeterminate form)

34. lim x→-3 (x^2+x-6)/(x^2-9) = 0/0 (indeterminate form)

35. lim x→-2 (x+2)/(x^2-x-6) = -4/0 (undefined)

36. lim x→5 (x-5)/(x^2-3x-10) = 0/0 (indeterminate form)

37. lim x→0 x/(x+3) - 1/3 = 0/3 - 1/3 = -1/3

38. lim x→0 x - 1/(x+2) + 1/2 = 0 - 1/2 + 1/2 = 0

39. lim x→25 (x-25)/(x-5) = 0/20 = 0

40. lim x→36 (x-36)/(x-6) = 0/30 = 0

41. lim h→0 h[(x+h)^2 - x^2] = h[2xh + h^2] = 2xh^2 + h^3 = 0

42. lim h→0 h[(x+h)^3 - x^3] = h[3x^2h + 3xh^2 + h^3] = 0

43. lim x→-1 7x - 1 = -7 - 1 = -8

44. lim x→-∞ (4x-5)/(8x+2) = -∞/-∞ (indeterminate form)

46. lim x→∞ (3x^2 + 2)/(x^2 + 2x - 5) = ∞/∞ (indeterminate form)

(a) lim x→3 f(x) = lim x→3 (x-1)/(2x+3