## Practice Problems

Work through the odd-numbered problems 1-29. Once you have completed the problem set, check your answers.

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