Practice Problems

Determine the limits in problems 1-15.

1. $\lim \limits_{x \rightarrow 1} \frac{x^{3}-1}{x^{2}-1}$

3. $\lim \limits_{x \rightarrow 0} \frac{\ln (1+3 x)}{5 x}$

5. $\lim \limits_{x \rightarrow 0} \frac{x \cdot e^{x}}{1-e^{x}}$

7. $\lim \limits_{x \rightarrow \infty} \frac{\ln (x)}{x}$

9. $\lim \limits_{x \rightarrow \infty} \frac{\ln (x)}{x^{p}}$ ($p$ is any positive number)

11. $\lim \limits_{x \rightarrow 0} \frac{1-\cos (3 x)}{x^{2}}$

13. $\lim \limits_{x \rightarrow a} \frac{x^{m}-a^{m}}{x^{n}-a^{n}}$

15. $\lim \limits_{x \rightarrow 0} \frac{1-\cos (x)}{x \cdot \cos (x)}$

17. Find a value for $p$ so $\lim \limits_{x \rightarrow 0} \frac{e^{p x}-1}{3 x}=5$.

19. (a) Evaluate $\lim \limits_{x \rightarrow \infty} \frac{e^{x}}{x}, \lim \limits_{x \rightarrow \infty} \frac{e^{x}}{x^{2}}, \lim \limits_{x \rightarrow \infty} \frac{e^{x}}{x^{5}}$.
(b) An algorithm is "exponential" if it requires $\mathrm{a} \cdot \mathrm{e}^{\mathrm{bn}}$ steps ($a$ and $\mathrm{b}$ are positive constants). An algorithm is "polynomial" if it requires $\mathrm{c} \cdot \mathrm{n}^{\mathrm{d}}$ steps ($c$ and $\mathrm{d}$ are positive constants). Show that polynomial algorithms require fewer steps than exponential algorithms for large problems.

Determine the limits in problems 21-29.

21. $\lim \limits_{x \rightarrow 0^{+}} \sin (x) \cdot \ln (x)$

23. $\lim \limits_{x \rightarrow 0^{+}} \sqrt{x} \cdot \ln (x)$

25. $\lim \limits_{x \rightarrow \infty}\left(1-3 / x^{2}\right)^{x}$

27. $\lim \limits_{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin (x)}\right)$

29. $\lim \limits_{x \rightarrow \infty}\left(\frac{x+5}{x}\right)^{1 / x}$

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2011/11/4-7LHopitalsRule.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. $3 / 2$

3. $3 / 5$

5. $-1$

7. $0$

9. $0$

11. $9 / 2$

13. For $a \neq 0: \frac{f^{\prime}}{g^{\prime}}=\frac{m}{n} x^{(m-n)} \rightarrow \frac{m}{n} a^{(m-n)}$
For $a=0,  \lim \limits_{x \rightarrow a} \frac{x^{m}-a^{m}}{x^{n}-a^{n}}= \begin{cases}0 & \text { if } m > n \\ 1 & \text { if } m=n \\ +\infty & \text { if } m < n \text { and }(m-n) \text { is even } \\ \text { DNE } & \text { if } m < n \text { and }(m-n) \text { is odd }\end{cases}$

15. $0$

17. $\frac{\mathrm{f}^{\prime}}{\mathrm{g}^{\prime}}=\frac{\mathrm{pe}^{\mathrm{px}}}{3} \rightarrow \frac{\mathrm{p}}{3}$ so $\mathrm{p}=3(5)=15$.

19. (a) All three limits are $+\infty$.

(b) After applying L'Hopital's Rule $d$ times, $\frac{\mathrm{f}^{(\mathrm{d})}}{\mathrm{g}^{(\mathrm{d})}}=\frac{\mathrm{a} \cdot \mathrm{b}^{\mathrm{n}} \cdot \mathrm{e}^{\text {bn }}}{\mathrm{c}(\mathrm{d})(\mathrm{d}-1)(\mathrm{d}-2) \ldots(2)(1)}=\frac{\text { constant } \cdot \mathrm{e}^{\mathrm{bn}}}{\text { another constant }} \rightarrow+\infty$.

21. $0$

23. $0$

25. $1$

27. $0$

29. $1$