Practice Problems

1. Fig. 9 shows $f(x)$ and $g(x)$ for $0 \leq x \leq 5$. Let $h(x)=\frac{f(x)}{g(x)}$.
(a) At what value of $x$ does $h(x)$ have a root?
(b) Determine the limits of $\mathrm{h}(\mathrm{x})$ as $\mathrm{x} \rightarrow 1^{+}, \mathrm{x} \rightarrow 1^{-}, \mathrm{x} \rightarrow 3^{+}$, and $\mathrm{x} \rightarrow 3^{-}$,
(c) Where does $\mathrm{h}(\mathrm{x})$ have a vertical asymptote?

Fig. 9

3. Fig. 11 shows $f(x)$ and $g(x)$ for $0 \leq x \leq 5$. Let $h(x)=\frac{f(x)}{g(x)}$, and determine the limits of $\mathrm{h}(\mathrm{x})$ as $\mathrm{x} \rightarrow 2^{+}, \mathrm{x} \rightarrow 2^{-}, \mathrm{x} \rightarrow 4^{+}$, and $\mathrm{x} \rightarrow 4^{-}$.

Fig. 11

For problems 5-23, calculate the limit of each expression as " $\mathbf{x} \rightarrow \infty$ ".

5. $\frac{28}{3 x-5}$

7. $\frac{4-3 x}{x+8}$

9. $\frac{\cos (3 x)}{5 x-1}$

11. $\frac{4+x \cdot \sin (x)}{2 x-3}$

13. $\frac{2 x^{2}-9}{3 x^{2}+10 x}$

15. $\frac{5 x^{2}-7 x+2}{2 x^{3}+4 x}$

17. $\frac{7 x^{2}+x \cdot \sin (x)}{3-x^{2}+\sin \left(7 x^{2}\right)}$

19. $\frac{\sqrt{9 x^{2}+16}}{2+\sqrt{x^{3}+1}}$

21. $\cos \left(\frac{7 x+4}{x^{2}+x+1}\right)$

23. $\ln (x+8)-\ln (x-5)$

25. Salt water with a concentration of $0.2$ pounds of salt per gallon flows into a large tank that initially contains $50$ gallons of pure water.
(a) If the flow rate of salt water into the tank is $4$ gallons per minute, what is the volume $V(t)$ of water and the amount $\mathrm{A}(\mathrm{t})$ of salt in the tank t minutes after the flow begins?
(b) Show that the salt concentration $C(t)$ at time $t$ is $C(t)=\frac{.8 t}{4 t+50}$.
(c) What happens to the concentration $\mathrm{C}(\mathrm{t})$ after a "long" time?
(d) Redo parts $(a) - (c)$ for a large tank which initially contains $200$ gallons of pure water.

For problems 27-41, calculate the limits.

27. $\lim \limits_{x \rightarrow 0} \frac{x+5}{x^{2}}$

29. $\lim \limits_{x \rightarrow 5} \frac{x-7}{(x-5)^{2}}$

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

33. $\lim \limits_{x \rightarrow 4^{+}} \frac{x+3}{4-x}$

35. $\lim \limits_{x \rightarrow 3^{+}} \frac{x^{2}-4}{x^{2}-2 x-3}$

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

39. $\lim \limits_{x \rightarrow 5} \frac{\sin (x-5)}{x-5}$

41. $\lim \limits_{x \rightarrow 0^{+}} \frac{1+\cos (x)}{1-e^{x}}$

In problems 43-49, write the equation of each asymptote for each function and state whether it is a vertical or horizontal asymptote.

43. $f(x)=\frac{x-3}{x^{2}}$

45. $f(x)=\frac{x+5}{x^{2}-4 x+3}$

47. $f(x)=\frac{x^{2}-4}{x^{2}+1}$

49. $f(x)=2+\frac{3-x}{x-1}$

In problems 51-59, write the equation of each asymptote for each function.

51. $f(x)=\frac{2 x^{2}+x+5}{x}$

53. $f(x)=\frac{1}{x-2}+\sin (x)$

55. $f(x)=x^{2}+\frac{x}{x^{2}+1}$

57. $f(x)=\frac{x \cdot \cos (x)}{x-3}$

59. $f(x)=\sqrt{\frac{x^{2}+3 x+2}{x+3}}$

1. (a) $\mathrm{h}$ has a root at $\mathrm{x}=1$.
(b) limits of $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x}) / \mathrm{g}(\mathrm{x})$: as $\mathrm{x} \rightarrow 1^{+}$ is $0$: as $\mathrm{x} \rightarrow 1^{-}$ is $0$: as $\mathrm{x} \rightarrow 3^{+}$ is $-\infty$: as $\mathrm{x} \rightarrow 3^{-}$ is $+\infty$
(c) $\mathrm{h}$ has a vertical asymptote at $\mathrm{x}=3$

3. limits of $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x}) / \mathrm{g}(\mathrm{x})$: as $\mathrm{x} \rightarrow 2^{+}$ is $+\infty$: as $\mathrm{x} \rightarrow 2^{-}$ is $-\infty$: as $\mathrm{x} \rightarrow 4^{+}$ is $0$: as $\mathrm{x} \rightarrow 4^{-}$ is $0$

5. $0$

7. $-3$

9. $0$

11. $DNE$

13. $2 / 3$

15. $0$

17. $-7$

19. $0$

21. $\cos (0)=1$

23. $\ln (1)=0$

25. (a) $\mathrm{V}(\mathrm{t})=50+4 \mathrm{t}$ gallons, and $\mathrm{A}(\mathrm{t})=0.8 \mathrm{t}$ pounds of salt
(b) $\mathrm{C}(\mathrm{t})=\frac{\text { amount of salt }}{\text { total amount of liquid }}=\frac{\mathrm{A}(\mathrm{t})}{\mathrm{V}(\mathrm{t})}=\frac{0.8 \mathrm{t}}{50+4 \mathrm{t}}$
(c) "after a long time" (as $\mathrm{t} \rightarrow \infty$), $\mathrm{C}(\mathrm{t}) \rightarrow 0.8 / 4=0.2$ pounds of salt per gallon.
(d) $\mathrm{V}(\mathrm{t})=200+4 \mathrm{t}, \mathrm{A}(\mathrm{t})=0.8 \mathrm{t}, \mathrm{C}(\mathrm{t})=\frac{0.8 \mathrm{t}}{200+4 \mathrm{t}} \rightarrow 0.8 / 4=0.2$ pounds of salt per gallon.

27. $+\infty$

29. $-\infty$

31. $-\infty$

33. $-\infty$

35. $+\infty$

37. $-\infty$

39. $1$

41. $-\infty$

43. Horizontal: $\mathrm{y}=0$. Vertical: $\mathrm{x}=0$.

45. Horizontal: $\mathrm{y}=0$. Vertical: $\mathrm{x}=3$ and $\mathrm{x}=1$

47. Horizontal: $\mathrm{y}=1$.

49. Horizontal: $\mathrm{y}=1$. Vertical: $\mathrm{x}=1$.

51. $\mathrm{y}=2 \mathrm{x}+1$. $\mathrm{x}=0$

53. $y=\sin (x)$. $x=2$

55. $\mathrm{y}=\mathrm{x}^{2}$

57. $\mathrm{y}=\cos (\mathrm{x})$. $\mathrm{x}=3$.

59. $\mathrm{y}=\sqrt{\mathrm{x}}$. $\mathrm{x}=-3$