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.

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

For the following exercises, refer to Figure 4.

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.