Problem Set: L’Hôpital’s Rule

For the following exercises (1-6), evaluate the limit.

1. Evaluate the limit limxexxlimxexx.

2. Evaluate the limit limxexxklimxexxk.

3. Evaluate the limit limxlnxxklimxlnxxk.

4. Evaluate the limit limxaxax2a2, a0limxaxax2a2, a0.

5. Evaluate the limit limxaxax3a3, a0limxaxax3a3, a0.

6. Evaluate the limit limxaxaxnan, a0limxaxaxnan, a0.

For the following exercises (7-11), determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

7. limx0+x2lnxlimx0+x2lnx

8. limxx1/xlimxx1/x

9. limx0x2/xlimx0x2/x

10. limx0x21/xlimx0x21/x

11. limxexxlimxexx

For the following exercises (12-40), evaluate the limits with either L’Hôpital’s rule or previously learned methods.

12. limx3x29x3limx3x29x3

13. limx3x29x+3limx3x29x+3

14. limx0(1+x)21xlimx0(1+x)21x

15. limxπ/2cosxπ/2xlimxπ/2cosxπ/2x

16. limxπxπsinxlimxπxπsinx

17. limx1x1sinxlimx1x1sinx

18. limx0(1+x)n1xlimx0(1+x)n1x

19. limx0(1+x)n1nxx2limx0(1+x)n1nxx2

20. limx0sinxtanxx3limx0sinxtanxx3

21. limx01+x1xxlimx01+x1xx

22. limx0exx1x2limx0exx1x2

23. limx0tanxxlimx0tanxx

24. limx1x1lnxlimx1x1lnx

25. limx0(x+1)1/xlimx0(x+1)1/x

26. limx1x3xx1limx1x3xx1

27. limx0+x2xlimx0+x2x

28. limxxsin(1x)limxxsin(1x)

29. limx0sinxxx2limx0sinxxx2

30. limx0+xln(x4)limx0+xln(x4)

31. limx(xex)limx(xex)

32. limxx2exlimxx2ex

33. limx03x2xxlimx03x2xx

34. limx01+1/x11/xlimx01+1/x11/x

35. limxπ/4(1tanx)cotxlimxπ/4(1tanx)cotx

36. limxxe1/xlimxxe1/x

37. limx0x1/cosxlimx0x1/cosx

38. limx0+x1/xlimx0+x1/x

39. limx0(11x)xlimx0(11x)x

40. limx(11x)xlimx(11x)x

For the following exercises (41-50), use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.

41. [T] limx0ex1xlimx0ex1x

42. [T] limx0xsin(1x)limx0xsin(1x)

43. [T] limx1x11cos(πx)limx1x11cos(πx)

44. [T] limx1ex11x1limx1ex11x1

45. [T] limx1(x1)2lnxlimx1(x1)2lnx

46. [T] limxπ1+cosxsinxlimxπ1+cosxsinx

47. [T] limx0(cscx1x)limx0(cscx1x)

48. [T] limx0+tan(xx)limx0+tan(xx)

49. [T] limx0+lnxsinxlimx0+lnxsinx

50. [T] limx0exexxlimx0exexx