3.4.2: Homework

Reading Questions

1. What are the two primary indeterminate forms for which L'Hôpital's Rule can be directly applied?
2. State L'Hôpital's Rule for the \(\frac{0}{0}\) case. What conditions must be met by the functions \(f\) and \(g\)?
3. If \(\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\), what limit do you evaluate according to L'Hôpital's Rule?
4. Does L'Hôpital's Rule apply if \(\displaystyle \lim_{x \to a} f(x) = 0\) but \(\lim_{x \to a} g(x) = 5\)? Why or why not?
5. How can the indeterminate form \(0 \cdot \infty\) be rewritten so that L'Hôpital's Rule might be applicable?
6. How can the indeterminate form \(\infty - \infty\) be rewritten so that L'Hôpital's Rule might be applicable?
7. What technique is used to handle indeterminate forms involving exponents, such as \(0^0\), \(\infty^0\), or \(1^\infty\), before L'Hôpital's Rule can be applied?
8. If \(\displaystyle \lim_{x \to a} \ln(y) = L\), what is \(\displaystyle \lim_{x \to a} y\)?
9. When comparing the growth rates of two functions \(f(x)\) and \(g(x)\) that both approach \(\infty\) as \(x \to \infty\), what does it mean if \(\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0\)? Which function grows more rapidly?
10. Based on the examples, which type of function generally grows faster as \(x \to \infty\): an exponential function (\(e^x\)), a power function (\(x^p, p>0\)), or a logarithmic function (\(\ln x\))?

Homework

In exercises 1 - 6, evaluate the limit.

1) Evaluate the limit \(\displaystyle \lim_{x \to \infty }\frac{e^x}{x}\).

2) Evaluate the limit \(\displaystyle \lim_{x \to \infty }\frac{e^x}{x^k}\).

Answer

\(\displaystyle \lim_{x \to \infty }\frac{e^x}{x^k} \quad = \quad \infty \)

3) Evaluate the limit \(\displaystyle \lim_{x \to \infty }\frac{\ln x}{x^k}\).

4) Evaluate the limit \(\displaystyle \lim_{x \to a}\frac{x−a}{x^2−a^2}\).

Answer

\(\displaystyle \lim_{x \to a}\frac{x−a}{x^2−a^2} \quad = \quad \frac{1}{2a}\)

5. Evaluate the limit \(\displaystyle \lim_{x \to a}\frac{x−a}{x^3−a^3}\).

6. Evaluate the limit \(\displaystyle \lim_{x \to a}\frac{x−a}{x^n−a^n}\).

Answer

\(\displaystyle \lim_{x \to a}\frac{x−a}{x^n−a^n} \quad = \quad \frac{1}{na^{n−1}}\)

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) \(\displaystyle \lim_{x \to 0^+}x^2\ln x\)

8) \(\displaystyle \lim_{x \to \infty }x^{1/x}\)

Answer

Cannot apply directly; use logarithms

9) \(\displaystyle \lim_{x \to 0}x^{2/x}\)

10) \(\displaystyle \lim_{x \to 0}\frac{x^2}{1/x}\)

Answer

Cannot apply directly; rewrite as \(\displaystyle \lim_{x \to 0}x^3\)

11) \(\displaystyle \lim_{x \to \infty }\frac{e^x}{x}\)

In exercises 12 - 40, evaluate the limits with either l'Hôpital's Rule or previously learned methods.

12) \(\displaystyle \lim_{x \to 3}\frac{x^2−9}{x−3}\)

Answer

\(\displaystyle \lim_{x \to 3}\frac{x^2−9}{x−3} \quad = \quad 6\)

13) \(\displaystyle \lim_{x \to 3}\frac{x^2−9}{x+3}\)

14) \(\displaystyle \lim_{x \to 0}\frac{(1+x)^{−2}−1}{x}\)

Answer

\(\displaystyle \lim_{x \to 0}\frac{(1+x)^{−2}−1}{x} \quad = \quad -2\)

15) \(\displaystyle \lim_{x \to \pi /2}\frac{\cos x}{\frac{ \pi }{2}−x}\)

16) \(\displaystyle \lim_{x \to \pi }\frac{x− \pi }{\sin x}\)

Answer

\(\displaystyle \lim_{x \to \pi }\frac{x− \pi }{\sin x} \quad = \quad -1\)

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

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

Answer

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

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

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

Answer

\(\displaystyle \lim_{x \to 0}\frac{\sin x−\tan x}{x^3} \quad = \quad −\frac{1}{2}\)

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

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

Answer

\(\displaystyle \lim_{x \to 0}\frac{e^x−x−1}{x^2} \quad = \quad \frac{1}{2}\)

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

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

Answer

\(\displaystyle \lim_{x \to 1}\frac{x-1}{\ln x} \quad = \quad 1\)

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

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

Answer

\(\displaystyle \lim_{x \to 1}\frac{\sqrt{x}−\sqrt[3]{x}}{x−1} \quad = \quad \frac{1}{6}\)

27) \(\displaystyle \lim_{x \to 0^+}x^{2x}\)

28) \(\displaystyle \lim_{x \to \infty }x\sin\left(\tfrac{1}{x}\right)\)

Answer

\(\displaystyle \lim_{x \to \infty }x\sin\left(\tfrac{1}{x}\right) \quad = \quad 1\)

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

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

Answer

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

31) \(\displaystyle \lim_{x \to \infty }(x−e^x)\)

32) \(\displaystyle \lim_{x \to \infty }x^2e^{−x}\)

Answer

\(\displaystyle \lim_{x \to \infty }x^2e^{−x} \quad = \quad 0\)

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

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

Answer

\(\displaystyle \lim_{x \to 0}\frac{1+1/x}{1−1/x} \quad = \quad -1\)

35) \(\displaystyle \lim_{x \to \pi /4}(1−\tan x)\cot x\)

36) \(\displaystyle \lim_{x \to \infty }xe^{1/x}\)

Answer

\(\displaystyle \lim_{x \to \infty }xe^{1/x} \quad = \quad \infty \)

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

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

Answer

\(\displaystyle \lim_{x \to 0^{+} }x^{1/x} \quad = \quad 0\)

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

40) \(\displaystyle \lim_{x \to \infty }\left(1−\frac{1}{x}\right)^x\)

Answer

\(\displaystyle \lim_{x \to \infty }\left(1−\frac{1}{x}\right)^x \quad = \quad \frac{1}{e}\)

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) [Technology Required] \(\displaystyle \lim_{x \to 0}\frac{e^x−1}{x}\)

42) [Technology Required] \(\displaystyle \lim_{x \to 0}x\sin\left(\tfrac{1}{x}\right)\)

Answer

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

43) [Technology Required] \(\displaystyle \lim_{x \to 1}\frac{x−1}{1−\cos( \pi x)}\)

44) [Technology Required] \(\displaystyle \lim_{x \to 1}\frac{e^{x−1}−1}{x−1}\)

Answer

\(\displaystyle \lim_{x \to 1}\frac{e^{x−1}−1}{x−1} \quad = \quad 1\)

45) [Technology Required] \(\displaystyle \lim_{x \to 1}\frac{(x−1)^2}{\ln x}\)

46) [Technology Required] \(\displaystyle \lim_{x \to \pi }\frac{1+\cos x}{\sin x}\)

Answer

\(\displaystyle \lim_{x \to \pi }\frac{1+\cos x}{\sin x} \quad = \quad 0\)

47) [Technology Required] \(\displaystyle \lim_{x \to 0}\left(\csc x−\frac{1}{x}\right)\)

48) [Technology Required] \(\displaystyle \lim_{x \to 0^+}\tan\left(x^x\right)\)

Answer

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

49) [Technology Required] \(\displaystyle \lim_{x \to 0^+}\frac{\ln x}{\sin x}\)

50) [Technology Required] \(\displaystyle \lim_{x \to 0}\frac{e^x−e^{−x}}{x}\)

Answer

\(\displaystyle \lim_{x \to 0}\frac{e^x−e^{−x}}{x} \quad = \quad 2\)