MA005: Calculus I

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. $1$