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Mathematics LibreTexts

4.7E: Exercises for Section 4.7

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In 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

In 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

In exercises 12 - 40, 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) \displaystyle \lim_{x→π/2}\frac{\cos x}{\frac{π}{2}−x}

16) \displaystyle \lim_{x→π}\frac{x−π}{\sin x}

Answer
\displaystyle \lim_{x→π}\frac{x−π}{\sin x} \quad = \quad -1

17) \displaystyle \lim_{x→1}\frac{x−1}{\sin x}

18) \displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x}

Answer
\displaystyle \lim_{x→0}\frac{(1+x)^n−1}{x} \quad = \quad n

19) \displaystyle \lim_{x→0}\frac{(1+x)^n−1−nx}{x^2}

20) \displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3}

Answer
\displaystyle \lim_{x→0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}

21) \displaystyle \lim_{x→0}\frac{\sqrt{1+x}−\sqrt{1−x}}{x}

22) \displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2}

Answer
\displaystyle \lim_{x→0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}

23) \displaystyle \lim_{x→0}\frac{\tan x}{\sqrt{x}}

24) \displaystyle \lim_{x→1}\frac{x-1}{\ln x}

Answer
\displaystyle \lim_{x→1}\frac{x-1}{\ln x} \quad = \quad 1

25) \displaystyle \lim_{x→0}\,(x+1)^{1/x}

26) \displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1}

Answer
\displaystyle \lim_{x→1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}

27) \displaystyle \lim_{x→0^+}x^{2x}

28) \displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right)

Answer
\displaystyle \lim_{x→∞}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1

29) \displaystyle \lim_{x→0}\frac{\sin x−x}{x^2}

30) \displaystyle \lim_{x→0^+}x\ln\left(x^4\right)

Answer
\displaystyle \lim_{x→0^+}x\ln\left(x^4\right) \quad = \quad 0

31) \displaystyle \lim_{x→∞}(x−e^x)

32) \displaystyle \lim_{x→∞}x^2e^{−x}

Answer
\displaystyle \lim_{x→∞}x^2e^{−x} \quad = \quad 0

33) \displaystyle \lim_{x→0}\frac{3^x−2^x}{x}

34) \displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x}

Answer
\displaystyle \lim_{x→0}\frac{1+1/x}{1−1/x} \quad = \quad -1

35) \displaystyle \lim_{x→π/4}(1−\tan x)\cot x

36) \displaystyle \lim_{x→∞}xe^{1/x}

Answer
\displaystyle \lim_{x→∞}xe^{1/x} \quad = \quad ∞

37) \displaystyle \lim_{x→0}x^{1/\cos x}

38) \displaystyle \lim_{x→0^{+} }x^{1/x}

Answer
\displaystyle \lim_{x→0^{+} }x^{1/x} \quad = \quad 0

39) \displaystyle \lim_{x→0}\left(1−\frac{1}{x}\right)^x

40) \displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x

Answer
\displaystyle \lim_{x→∞}\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}

For 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] \displaystyle \lim_{x→0}\frac{e^x−1}{x}

42) [T] \displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right)

Answer
\displaystyle \lim_{x→0}x\sin\left(\tfrac{1}{x}\right) \quad = \quad 0

43) [T] \displaystyle \lim_{x→1}\frac{x−1}{1−\cos(πx)}

44) [T] \displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1}

Answer
\displaystyle \lim_{x→1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1

45) [T] \displaystyle \lim_{x→1}\frac{(x−1)^2}{\ln x}

46) [T] \displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x}

Answer
\displaystyle \lim_{x→π}\frac{1+\cos x}{\sin x} \quad = \quad 0

47) [T] \displaystyle \lim_{x→0}\left(\csc x−\frac{1}{x}\right)

48) [T] \displaystyle \lim_{x→0^+}\tan\left(x^x\right)

Answer
\displaystyle \lim_{x→0^+}\tan\left(x^x\right) \quad = \quad \tan 1

49) [T] \displaystyle \lim_{x→0^+}\frac{\ln x}{\sin x}

50) [T] \displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x}

Answer
\displaystyle \lim_{x→0}\frac{e^x−e^{−x}}{x} \quad = \quad 2

4.7E: Exercises for Section 4.7 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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