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7.7E: Exercises for L'Hôpital's Rule

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Terms and Concepts

1. List the different indeterminate forms described in this section.

Answer:
The forms 0,,1,0, and 00 are all considered indeterminate.

2. List similar looking forms that are not indeterminate.

Answer:
Among others, the forms ,+,,0,10, and are not considered indeterminate, as these limits can be determined clearly.

3. T/F: l'Hôpital's Rule states that ddx(f(x)g(x))=f(x)g(x).

Answer:
False. L'Hôpital's Rule is a method for taking limits of rational functions in certain cases. It does not replace the Quotient Rule when taking the derivative of these rational functions.

4. Explain what the indeterminate form "1" means. Why is it indeterminate?

Answer:
When a limit has the form "1", this means that the function in the base of the exponent is approaching 1, while the function in the exponent is approaching . It's indeterminate, since if the base function is approaching 1, but always is less than 1, then the limit could be 0, while if the base function were approaching 1, but always is greater than 1, the limit could be . But since this uncertainty exists, the limit could, in fact, be anything.

5. Explain why limits of the form are indeterminate.

Answer:
Limits with this form depend on the relative speed with which the two terms are approaching . If the first term approaches faster than the second term, the limit would be . If the second term approaches faster than the first term, the limit would be . But if they both appraoch at about the same rates, the limit could be anything!

6. Fill in the blanks" The Quotient Rule is applied to f(x)g(x) when taking its _____; l'Hôpital's Rule is applied when taking _______ of f(x)g(x) when the form is _______ or _______.

Answer:
derivative; limits; 00 or ±±

7. Create (but do not evaluate) a limit that initially has the form "0".

8. Create a function f(x) such that limx1f(x) initially has the form "00".

Problems

For exercises 1 - 6, evaluate the limit.

1) Evaluate the limit limxexx.

2) Evaluate the limit limxexxk.

Answer:
limxexxk=

3) Evaluate the limit limxlnxxk.

4) Evaluate the limit limxaxax2a2.

Answer:
limxaxax2a2=12a

5. Evaluate the limit limxaxax3a3.

6. Evaluate the limit limxaxaxnan.

Answer:
limxaxaxnan=1nan1

For 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+x2lnx

8) limxx1/x

Answer:
Cannot apply directly; use logarithms

9) limx0x2/x

10) limx0x21/x

Answer:
Cannot apply directly; rewrite as limx0x3

11) limxexx

For exercises 12 - 44, evaluate the limits with either L’Hôpital’s rule or previously learned methods.

12) limx3x29x3

Answer:
limx3x29x3=6

13) limx3x29x+3

14) limx0(1+x)21x

Answer:
limx0(1+x)21x=2

15) limxπ/2cosxπ2x

16) limxπxπsinx

Answer:
limxπxπsinx=1

17) limx1x1sinx

18) limx0(1+x)n1x

Answer:
limx0(1+x)n1x=n

19) limx0(1+x)n1nxx2

20) limx0sinxtanxx3

Answer:
limx0sinxtanxx3=12

21) limx01+x1xx

22) limx0exx1x2

Answer:
limx0exx1x2=12

23) limx0tanxx

24) limx1x1lnx

Answer:
limx1x1lnx=1

25) limx0(x+1)1/x

26) limx1x3xx1

Answer:
limx1x3xx1=16

27) limx0+x2x

28) limxxsin(1x)

Answer:
limxxsin(1x)=1

29) limx0sinxxx2

30) limx0+xln(x4)

Answer:
limx0+xln(x4)=0

31) limx(xex)

32) limxx2ex

Answer:
limxx2ex=0

33) limx1+[1lnx11x]

34) limx3+[5x29xx3]

Answer:
limx3+[5x29xx3]=limx3+5x23xx29=

35) limx2x23x+2

Note:
L’Hôpital’s rule fails to help us find this limit, although the form seems appropriate. But you can evaluate this limit using techniques you learned earlier in calculus.

36) limx(x+7x+3)x

Answer:
limx(x+7x+3)x=e4

37) limx03x2xx

38) limx01+1/x11/x

Answer:
limx01+1/x11/x=1

39) limxπ/4(1tanx)cotx

40) limxxe1/x

Answer:
limxxe1/x=

41) limx0x1/cosx

42) limx0+x1/x

Answer:
limx0+x1/x=0

43) limx0(11x)x

44) limx(11x)x

Answer:
limx(11x)x=1e

For exercises 45 - 54, 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.

45) [T] limx0ex1x

46) [T] limx0xsin(1x)

Answer:
limx0xsin(1x)=0

47) [T] limx1x11cos(πx)

48) [T] limx1ex11x1

Answer:
limx1ex11x1=1

49) [T] limx1(x1)2lnx

50) [T] limxπ1+cosxsinx

Answer:
limxπ1+cosxsinx=0

51) [T] limx0(cscx1x)

52) [T] limx0+tan(xx)

Answer:
limx0+tan(xx)=tan1

53) [T] limx0+lnxsinx

54) [T] limx0exexx

Answer:
limx0exexx=2

 

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/

  • Terms and Concepts 1-4, and 6-8 were adapted from Apex Calculus by Paul Seeburger (Monroe Community College)
  • Terms and Concepts Problem 5 was created by Paul Seeburger
  • Problems 33 and 34 from Apex Calculus. Solution to 34 by Paul Seeburger
  • Problems 35 and 36 and solutions for Terms and Concepts Problems 1-6 and for Problem 36 were added by Paul Seeburger

7.7E: Exercises for L'Hôpital's Rule is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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