## Problem Set 71: Finding Limits: Properties of Limits

1. Give an example of a type of function $f$ whose limit, as $x$ approaches $a$, is $f\left(a\right)$.

2. When direct substitution is used to evaluate the limit of a rational function as $x$ approaches $a$ and the result is $f\left(a\right)=\frac{0}{0}$, does this mean that the limit of $f$ does not exist?

3. What does it mean to say the limit of $f\left(x\right)$, as $x$ approaches $c$, is undefined?

For the following exercises, evaluate the limits algebraically.

4. $\underset{x\to 0}{\mathrm{lim}}\left(3\right)$

5. $\underset{x\to 2}{\mathrm{lim}}\left(\dfrac{-5x}{{x}^{2}-1}\right)$

6. $\underset{x\to 2}{\mathrm{lim}}\left(\dfrac{{x}^{2}-5x+6}{x+2}\right)$

7. $\underset{x\to 3}{\mathrm{lim}}\left(\dfrac{{x}^{2}-9}{x - 3}\right)$

8. $\underset{x\to -1}{\mathrm{lim}}\left(\dfrac{{x}^{2}-2x - 3}{x+1}\right)$

9. $\underset{x\to \frac{3}{2}}{\mathrm{lim}}\left(\dfrac{6{x}^{2}-17x+12}{2x - 3}\right)$

10. $\underset{x\to -\frac{7}{2}}{\mathrm{lim}}\left(\dfrac{8{x}^{2}+18x - 35}{2x+7}\right)$

11. $\underset{x\to 3}{\mathrm{lim}}\left(\dfrac{{x}^{2}-9}{x - 5x+6}\right)$

12. $\underset{x\to -3}{\mathrm{lim}}\left(\dfrac{-7{x}^{4}-21{x}^{3}}{-12{x}^{4}+108{x}^{2}}\right)$

13. $\underset{x\to 3}{\mathrm{lim}}\left(\dfrac{{x}^{2}+2x - 3}{x - 3}\right)$

14. $\underset{h\to 0}{\mathrm{lim}}\left(\dfrac{{\left(3+h\right)}^{3}-27}{h}\right)$

15. $\underset{h\to 0}{\mathrm{lim}}\left(\dfrac{{\left(2-h\right)}^{3}-8}{h}\right)$

16. $\underset{h\to 0}{\mathrm{lim}}\left(\dfrac{{\left(h+3\right)}^{2}-9}{h}\right)$

17. $\underset{h\to 0}{\mathrm{lim}}\left(\dfrac{\sqrt{5-h}-\sqrt{5}}{h}\right)$

18. $\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{\sqrt{3-x}-\sqrt{3}}{x}\right)$

19. $\underset{x\to 9}{\mathrm{lim}}\left(\dfrac{{x}^{2}-81}{3-\sqrt{x}}\right)$

20. $\underset{x\to 1}{\mathrm{lim}}\left(\dfrac{\sqrt{x}-{x}^{2}}{1-\sqrt{x}}\right)$

21. $\underset{x\to 0}{\mathrm{lim}}\left(\dfrac{x}{\sqrt{1+2x}-1}\right)$

22. $\underset{x\to \frac{1}{2}}{\mathrm{lim}}\left(\dfrac{{x}^{2}-\frac{1}{4}}{2x - 1}\right)$

23. $\underset{x\to 4}{\mathrm{lim}}\left(\dfrac{{x}^{3}-64}{{x}^{2}-16}\right)$

24. $\underset{x\to {2}^{-}}{\mathrm{lim}}\left(\dfrac{|x - 2|}{x - 2}\right)$

25. $\underset{x\to {2}^{+}}{\mathrm{lim}}\left(\dfrac{|x - 2|}{x - 2}\right)$

26. $\underset{x\to 2}{\mathrm{lim}}\left(\dfrac{|x - 2|}{x - 2}\right)$

27. $\underset{x\to {4}^{-}}{\mathrm{lim}}\left(\dfrac{|x - 4|}{4-x}\right)$

28. $\underset{x\to {4}^{+}}{\mathrm{lim}}\left(\dfrac{|x - 4|}{4-x}\right)$

29. $\underset{x\to 4}{\mathrm{lim}}\left(\dfrac{|x - 4|}{4-x}\right)$

30. $\underset{x\to 2}{\mathrm{lim}}\left(\dfrac{-8+6x-{x}^{2}}{x - 2}\right)$

For the following exercise, use the given information to evaluate the limits: $\underset{x\to c}{\mathrm{lim}}f\left(x\right)=3$, $\underset{x\to c}{\mathrm{lim}}g\left(x\right)=5$

31. $\underset{x\to c}{\mathrm{lim}}\left[2f\left(x\right)+\sqrt{g\left(x\right)}\right]$

32. $\underset{x\to c}{\mathrm{lim}}\left[3f\left(x\right)+\sqrt{g\left(x\right)}\right]$

33. $\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}$

For the following exercises, evaluate the following limits.

34. $\underset{x\to 2}{\mathrm{lim}}\cos \left(\pi x\right)$

35. $\underset{x\to 2}{\mathrm{lim}}\sin \left(\pi x\right)$

36. $\underset{x\to 2}{\mathrm{lim}}\sin \left(\frac{\pi }{x}\right)$

37. ${f}\left(x\right)=\begin{cases}2x^{2}+2x+1, \hfill& x\leq0 \\ x-3, \hfill& x>0\end{cases};\underset{x\to 0^{+}}{\mathrm{lim}}f \left(x\right)$

38. ${f}\left(x\right)=\begin{cases}2x^{2}+2x+1, \hfill& x\leq0 \\ x-3, \hfill& x>0\end{cases};\underset{x\to 0^{-}}{\mathrm{lim}}f \left(x\right)$

39. ${f}\left(x\right)=\begin{cases}2x^{2}+2x+1, \hfill& x\leq0 \\ x-3, \hfill& x>0\end{cases};\underset{x\to 0}{\mathrm{lim}}f \left(x\right)$

40. $\underset{x\to 4}{\mathrm{lim}}\dfrac{\sqrt{x+5}-3}{x - 4}$

41. $\underset{x\to {3}^{+}}{\mathrm{lim}}\dfrac{{x}^{2}}{{x}^{2}-9}$

For the following exercises, find the average rate of change $\frac{f\left(x+h\right)-f\left(x\right)}{h}$.

42. $f\left(x\right)=x+1$

43. $f\left(x\right)=2{x}^{2}-1$

44. $f\left(x\right)={x}^{2}+3x+4$

45. $f\left(x\right)={x}^{2}+4x - 100$

46. $f\left(x\right)=3{x}^{2}+1$

47. $f\left(x\right)=\cos \left(x\right)$

48. $f\left(x\right)=2{x}^{3}-4x$

49. $f\left(x\right)=\frac{1}{x}$

50. $f\left(x\right)=\frac{1}{{x}^{2}}$

51. $f\left(x\right)=\sqrt{x}$

52. Find an equation that could be represented by Figure 2.

Figure 2

53. Find an equation that could be represented by Figure 3.

Figure 4

For the following exercises, refer to Figure 4.

Figure 5

54. What is the right-hand limit of the function as $x$ approaches 0?

55. What is the left-hand limit of the function as $x$ approaches 0?

56. The position function $s\left(t\right)=-16{t}^{2}+144t$ gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval $\left[1,2\right]$ .

57. The height of a projectile is given by $s\left(t\right)=-64{t}^{2}+192t$ Find the average rate of change of the height from $t=1$ second to $t=1.5$ seconds.

58. The amount of money in an account after $t$ years compounded continuously at 4.25% interest is given by the formula $A={A}_{0}{e}^{0.0425t}$, where ${A}_{0}$ is the initial amount invested. Find the average rate of change of the balance of the account from $t=1$ year to $t=2$ years if the initial amount invested is \$1,000.00.