L'Hopital's rule i

L'Hopital's Rule

1. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

2. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

3. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate type. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

4. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

5. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

6. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  Since lim sin(x) = 0 and lim x = 0, x  x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

7. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's rule: x  x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

8. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim sin(x) x = lim [sin(x)]' [x]' x  x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

9. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim sin(x) x = lim [sin(x)]' [x]' x  x  = lim cos(x) 1 x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

10. L'Hopital's Rule: Given lim f(x) = lim g(x) = 0, then then lim f(x) g(x) = lim f '(x) g'(x) if the limit exists, or its ± ∞ . x  a x  a x  a x  a We call this type of limit as the " 0 " indeterminate L'Hopital's rule passes the calculation of the " 0 " form to the calculation of derivatives. Example: A. Find lim sin(x) x x  Since lim sin(x) = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim sin(x) x = lim [sin(x)]' [x]' x  x  = lim cos(x) 1 x  = 1. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) type.

11. B. Find lim e x – 1 x x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

12. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

13. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

14. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

15. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

16. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. C. Find lim e x – 1 x 2 x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

17. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. C. Find lim e x – 1 x 2 x  + Since lim e x – 1 = 0 and lim x 2 = 0, use the L'Hopital's rule: x  + x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

18. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. C. Find lim e x – 1 x 2 x  + Since lim e x – 1 = 0 and lim x 2 = 0, use the L'Hopital's rule: lim e x – 1 x 2 = lim [e x – 1 ]' [x 2 ]' x  + x  + x  + x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

19. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. C. Find lim e x – 1 x 2 x  + Since lim e x – 1 = 0 and lim x 2 = 0, use the L'Hopital's rule: lim e x – 1 x 2 = lim [e x – 1 ]' [x 2 ]' x  + = lim e x 2x x  + x  + x  + x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

20. B. Find lim e x – 1 x x  Since lim e x – 1 = 0 and lim x = 0, use the L'Hopital's rule: x  x  lim e x – 1 x = lim [e x – 1 ]' [x]' x  x  = lim e x 1 x  = 1. C. Find lim e x – 1 x 2 x  + Since lim e x – 1 = 0 and lim x 2 = 0, use the L'Hopital's rule: lim e x – 1 x 2 = lim [e x – 1 ]' [x 2 ]' x  + = lim e x 2x x  + = ∞ . x  + x  + x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

21. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

22. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . 1/x sin(1/x 2 ) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

23. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: sin(1/x 2 ) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

24. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: lim 1 x(sin(1/x 2 )) x  + ∞ sin(1/x 2 ) = lim 1/x sin(1/x 2 ) x  + ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

25. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: lim 1 x(sin(1/x 2 )) x  + ∞ sin(1/x 2 ) = lim 1/x sin(1/x 2 ) x  + ∞ = lim [1/x]' [sin(1/x 2 )]' x  + ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

26. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: lim 1 x(sin(1/x 2 )) x  + ∞ sin(1/x 2 ) = lim 1/x sin(1/x 2 ) x  + ∞ = lim [1/x]' [sin(1/x 2 )]' x  + ∞ = lim -x -2 cos(1/x 2 )(-2x -3 ) x  + ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

27. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: lim 1 x(sin(1/x 2 )) x  + ∞ sin(1/x 2 ) = lim 1/x sin(1/x 2 ) x  + ∞ = lim [1/x]' [sin(1/x 2 )]' x  + ∞ = lim -x -2 cos(1/x 2 )(-2x -3 ) x  + ∞ = lim x 2cos(1/x 2 ) x  + ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

28. D. Find lim 1 x(sin(1/x 2 )) x  + ∞ Put the limit in the form of . lim 1/x x  + ∞ 1/x = lim x  + ∞ sin(1/x 2 ) = 0 , use the L'Hopital's rule: lim 1 x(sin(1/x 2 )) x  + ∞ sin(1/x 2 ) = lim 1/x sin(1/x 2 ) x  + ∞ = lim [1/x]' [sin(1/x 2 )]' x  + ∞ = lim -x -2 cos(1/x 2 )(-2x -3 ) x  + ∞ = lim x 2cos(1/x 2 ) x  + ∞ = ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

29. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

30. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

31. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 = lim [2x]' [3x + 2]' x  L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

32. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 = lim [2x]' [3x + 2]' x  = 2 3 . L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

33. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 = lim [2x]' [3x + 2]' x  = 2 3 . are of the " ∞ " indeterminate forms. ∞ If lim f(x) = lim g(x) = ∞, then lim x  a x  a f(x) g(x) x  a L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

34. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 = lim [2x]' [3x + 2]' x  = 2 3 . are of the " ∞ " indeterminate forms. L'Hopital’s Rule applies ∞ If lim f(x) = lim g(x) = ∞, then lim x  a x  a f(x) g(x) x  a in these situations also, L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

35. Remark: We need lim f(x) = lim g(x) = 0 for L'Hopital's Rule (for the 0/0-form). x  a x  a E. lim 2x 3x + 2 x  Example: = 0 = lim [2x]' [3x + 2]' x  = 2 3 . are of the " ∞ " indeterminate forms. L'Hopital’s Rule applies ∞ If lim f(x) = lim g(x) = ∞, then lim x  a x  a f(x) g(x) x  a in these situations also, lim f(x) g(x) = lim f '(x) g'(x). x  a x  a that is L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

36. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

37. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

38. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ Hence 3x 2 – 4 2x 2 + x – 5 lim x  ∞ = [3x 2 – 4]' [2x 2 + x – 5]' lim x  ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

39. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ Hence 3x 2 – 4 2x 2 + x – 5 lim x  ∞ = [3x 2 – 4]' [2x 2 + x – 5]' lim x  ∞ 6x 4x + 1 lim x  ∞ = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

40. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ Hence 3x 2 – 4 2x 2 + x – 5 lim x  ∞ = [3x 2 – 4]' [2x 2 + x – 5]' lim x  ∞ 6x 4x + 1 lim x  ∞ = use L'Hopital's Rule again L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

41. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ Hence 3x 2 – 4 2x 2 + x – 5 lim x  ∞ = [3x 2 – 4]' [2x 2 + x – 5]' lim x  ∞ 6x 4x + 1 lim x  ∞ = = use L'Hopital's Rule again [6x]' [4x + 1]' lim x  ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

42. Example: F. Find lim 3x 2 – 4 2x 2 + x – 5 x  ∞ It’s the form. " ∞ " ∞ Hence 3x 2 – 4 2x 2 + x – 5 lim x  ∞ = [3x 2 – 4]' [2x 2 + x – 5]' lim x  ∞ 6x 4x + 1 lim x  ∞ = = use L'Hopital's Rule again [6x]' [4x + 1]' lim x  ∞ = 6 4 L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) = 3 2

43. G. e x P(x) x  ∞ where p(x) is a polynomial. Find lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

44. G. e x P(x) x  ∞ where p(x) is a polynomial Find lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) of degree > 0 .

45. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ e x P(x) [e x ]' [P(x)]' L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

46. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ e x P(x) [e x ]' [P(x)]' = lim x  ∞ e x [P(x)]' L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

47. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ e x P(x) [e x ]' [P(x)]' = lim x  ∞ e x [P(x)]' Use the L'Hopital's Rule repeatedly if necessary , eventually the denominator is a non-zero constant K. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

48. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ = lim x  ∞ e x K e x P(x) [e x ]' [P(x)]' = lim x  ∞ e x [P(x)]' Use the L'Hopital's Rule repeatedly if necessary , eventually the denominator is a non-zero constant K. Hence lim x  ∞ e x P(x) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

49. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ = lim x  ∞ e x K = ∞ e x P(x) [e x ]' [P(x)]' = lim x  ∞ e x [P(x)]' Use the L'Hopital's Rule repeatedly if necessary , eventually the denominator is a non-zero constant K. Hence lim x  ∞ e x P(x) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

50. It’s the form. " ∞ " ∞ Hence lim x  ∞ = lim x  ∞ = lim x  ∞ e x K We say that e x goes to ∞ faster than any polynomial. = ∞ e x P(x) [e x ]' [P(x)]' = lim x  ∞ e x [P(x)]' Use the L'Hopital's Rule repeatedly if necessary , eventually the denominator is a non-zero constant K. Hence lim x  ∞ e x P(x) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) G. e x P(x) x  ∞ where p(x) is a polynomial Find lim of degree > 0 .

51. H. x p Ln(x) x  ∞ where p > 0. Find lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

52. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ where p > 0. Find lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

53. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ x p Ln(x) [x p ]' [Ln(x)]' L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

54. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ = lim x  ∞ px p-1 x -1 x p Ln(x) [x p ]' [Ln(x)]' L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

55. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ = lim x  ∞ px p-1 x -1 x p Ln(x) [x p ]' [Ln(x)]' = lim x  ∞ px p L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

56. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ = lim x  ∞ px p-1 x -1 x p Ln(x) [x p ]' [Ln(x)]' = lim x  ∞ px p = ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

57. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ = lim x  ∞ px p-1 x -1 We say that x p (p > 0) goes to ∞ faster than Ln(x). x p Ln(x) [x p ]' [Ln(x)]' = lim x  ∞ px p = ∞ L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

58. H. x p Ln(x) x  ∞ It’s the form. " ∞ " ∞ Hence lim x  ∞ = where p > 0. Find lim lim x  ∞ = lim x  ∞ px p-1 x -1 We say that x p (p > 0) goes to ∞ faster than Ln(x). x p Ln(x) [x p ]' [Ln(x)]' = lim x  ∞ px p = ∞ The ∞/∞ form is the same as ∞ * form since ∞/∞ = ∞ * ( 1/ ∞) and we may view that 1/ ∞ a s 0. L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

59. I. x  + Find lim sin(x)Ln(x) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

60. I. x  + Find lim sin(x)Ln(x) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

61. I. x  + Find lim sin(x)Ln(x) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as 1/sin(x) Ln(x) L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

62. I. x  + in the form. ∞ ∞ Find lim sin(x)Ln(x) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

63. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim Find lim sin(x)Ln(x) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

64. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

65. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

66. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + = lim x - sin(x)tan(x) x  + 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

67. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + = lim x - sin(x)tan(x) x  + = x -sin(x) lim x  + tan(x) lim x  + 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

68. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + = lim x - sin(x)tan(x) x  + = x -sin(x) lim x  + tan(x) lim x  + 1 1/sin(x) Ln(x) = L'Hopital's Rule (0/0, ∞/∞, ∞ * forms)

69. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + = lim x - sin(x)tan(x) x  + = x -sin(x) lim x  + tan(x) lim x  + -1 1/sin(x) Ln(x) =

70. I. x  + in the form. ∞ ∞ Hence lim sin(x)Ln(x) = lim = Find lim sin(x)Ln(x) lim L'Hopital's Rule (0/0, ∞/∞, ∞ * forms) Lim sin(x) = 0 and lim Ln(x) = - ∞ so its the 0* ∞ form. x  + x  + Write it as csc(x) Ln(x) csc(x) Ln(x) x  + x  + [csc(x)]' [Ln(x)]' x  + = lim -csc(x)cot(x) 1/x x  + = lim x - sin(x)tan(x) x  + = x -sin(x) lim x  + tan(x) = 0 lim x  + -1 1/sin(x) Ln(x) =