For the following exercises (1-6), evaluate the limit.
1. Evaluate the limit limx→∞exxlimx→∞exx.
2. Evaluate the limit limx→∞exxklimx→∞exxk.
3. Evaluate the limit limx→∞lnxxklimx→∞lnxxk.
4. Evaluate the limit limx→ax−ax2−a2, a≠0limx→ax−ax2−a2, a≠0.
5. Evaluate the limit limx→ax−ax3−a3, a≠0limx→ax−ax3−a3, a≠0.
6. Evaluate the limit limx→ax−axn−an, a≠0limx→ax−axn−an, a≠0.
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. limx→0+x2lnxlimx→0+x2lnx
8. limx→∞x1/xlimx→∞x1/x
9. limx→0x2/xlimx→0x2/x
10. limx→0x21/xlimx→0x21/x
11. limx→∞exxlimx→∞exx
For the following exercises (12-40), evaluate the limits with either L’Hôpital’s rule or previously learned methods.
12. limx→3x2−9x−3limx→3x2−9x−3
13. limx→3x2−9x+3limx→3x2−9x+3
14. limx→0(1+x)−2−1xlimx→0(1+x)−2−1x
15. limx→π/2cosxπ/2−xlimx→π/2cosxπ/2−x
16. limx→πx−πsinxlimx→πx−πsinx
17. limx→1x−1sinxlimx→1x−1sinx
18. limx→0(1+x)n−1xlimx→0(1+x)n−1x
19. limx→0(1+x)n−1−nxx2limx→0(1+x)n−1−nxx2
20. limx→0sinx−tanxx3limx→0sinx−tanxx3
21. limx→0√1+x−√1−xxlimx→0√1+x−√1−xx
22. limx→0ex−x−1x2limx→0ex−x−1x2
23. limx→0tanx√xlimx→0tanx√x
24. limx→1x−1lnxlimx→1x−1lnx
25. limx→0(x+1)1/xlimx→0(x+1)1/x
26. limx→1√x−3√xx−1limx→1√x−3√xx−1
27. limx→0+x2xlimx→0+x2x
28. limx→∞xsin(1x)limx→∞xsin(1x)
29. limx→0sinx−xx2limx→0sinx−xx2
30. limx→0+xln(x4)limx→0+xln(x4)
31. limx→∞(x−ex)limx→∞(x−ex)
32. limx→∞x2e−xlimx→∞x2e−x
33. limx→03x−2xxlimx→03x−2xx
34. limx→01+1/x1−1/xlimx→01+1/x1−1/x
35. limx→π/4(1−tanx)cotxlimx→π/4(1−tanx)cotx
36. limx→∞xe1/xlimx→∞xe1/x
37. limx→0x1/cosxlimx→0x1/cosx
38. limx→0+x1/xlimx→0+x1/x
39. limx→0(1−1x)xlimx→0(1−1x)x
40. limx→∞(1−1x)xlimx→∞(1−1x)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] limx→0ex−1xlimx→0ex−1x
42. [T] limx→0xsin(1x)limx→0xsin(1x)
43. [T] limx→1x−11−cos(πx)limx→1x−11−cos(πx)
44. [T] limx→1ex−1−1x−1limx→1ex−1−1x−1
45. [T] limx→1(x−1)2lnxlimx→1(x−1)2lnx
46. [T] limx→π1+cosxsinxlimx→π1+cosxsinx
47. [T] limx→0(cscx−1x)limx→0(cscx−1x)
48. [T] limx→0+tan(xx)limx→0+tan(xx)
49. [T] limx→0+lnxsinxlimx→0+lnxsinx
50. [T] limx→0ex−e−xxlimx→0ex−e−xx
Candela Citations
- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction