Practice Problems

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
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