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}}