1. Explain the difference between a value at [latex]x=a[/latex] and the limit as [latex]x[/latex] approaches [latex]a[/latex].

2. Explain why we say a function does not have a limit as [latex]x[/latex] approaches [latex]a[/latex] if, as [latex]x[/latex] approaches [latex]a[/latex], the left-hand limit is not equal to the right-hand limit.

For the following exercises, estimate the functional values and the limits from the graph of the function [latex]f[/latex] provided in Figure 14.

Figure 14

3. [latex]\underset{x\to -{2}^{-}}{\mathrm{lim}}f\left(x\right)[/latex]

4. [latex]\underset{x\to -{2}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]

5. [latex]\underset{x\to -2}{\mathrm{lim}}f\left(x\right)[/latex]

6. [latex]f\left(-2\right)[/latex]

7. [latex]\underset{x\to -{1}^{-}}{\mathrm{lim}}f\left(x\right)[/latex]

8. [latex]\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]

9. [latex]\underset{x\to 1}{\mathrm{lim}}f\left(x\right)[/latex]

10. [latex]f\left(1\right)[/latex]

11. [latex]\underset{x\to {4}^{-}}{\mathrm{lim}}f\left(x\right)[/latex]

12. [latex]\underset{x\to {4}^{+}}{\mathrm{lim}}f\left(x\right)[/latex]

13. [latex]\underset{x\to 4}{\mathrm{lim}}f\left(x\right)[/latex]

14. [latex]f\left(4\right)[/latex]

For the following exercises, draw the graph of a function from the functional values and limits provided.

15. [latex]\underset{x\to {0}^{-}}{\mathrm{lim}}f\left(x\right)=2,\underset{x\to {0}^{+}}{\mathrm{lim}}f\left(x\right)=-3,\underset{x\to 2}{\mathrm{lim}}f\left(x\right)=2,f\left(0\right)=4,f\left(2\right)=-1,f\left(-3\right)\text{ does not exist}[/latex].

16. [latex]\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=0,\underset{x\to {2}^{+}}{\mathrm{lim}}=-2,\underset{x\to 0}{\mathrm{lim}}f\left(x\right)=3,f\left(2\right)=5,f\left(0\right)[/latex]

17. [latex]\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=2,\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=-3,\underset{x\to 0}{\mathrm{lim}}f\left(x\right)=5,f\left(0\right)=1,f\left(1\right)=0[/latex]

18. [latex]\underset{x\to {3}^{-}}{\mathrm{lim}}f\left(x\right)=0,\underset{x\to {3}^{+}}{\mathrm{lim}}f\left(x\right)=5,\underset{x\to 5}{\mathrm{lim}}f\left(x\right)=0,f\left(5\right)=4,f\left(3\right)\text{ does not exist}[/latex].

19. [latex]\underset{x\to 4}{\mathrm{lim}}f\left(x\right)=6,\underset{x\to {6}^{+}}{\mathrm{lim}}f\left(x\right)=-1,\underset{x\to 0}{\mathrm{lim}}f\left(x\right)=5,f\left(4\right)=6,f\left(2\right)=6[/latex]

20. [latex]\underset{x\to -3}{\mathrm{lim}}f\left(x\right)=2,\underset{x\to {1}^{+}}{\mathrm{lim}}f\left(x\right)=-2,\underset{x\to 3}{\mathrm{lim}}f\left(x\right)=-4,f\left(-3\right)=0,f\left(0\right)=0[/latex]

21. [latex]\underset{x\to \pi }{\mathrm{lim}}f\left(x\right)={\pi }^{2},\underset{x\to -\pi }{\mathrm{lim}}f\left(x\right)=\frac{\pi }{2},\underset{x\to {1}^{-}}{\mathrm{lim}}f\left(x\right)=0,f\left(\pi \right)=\sqrt{2},f\left(0\right)\text{ does not exist}[/latex].

For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as [latex]x[/latex] approaches 0.

22. [latex]f\left(x\right)={\left(1+x\right)}^{\frac{1}{x}}[/latex]

23. [latex]g\left(x\right)={\left(1+x\right)}^{\frac{2}{x}}[/latex]

24. [latex]h\left(x\right)={\left(1+x\right)}^{\frac{3}{x}}[/latex]

25. [latex]i\left(x\right)={\left(1+x\right)}^{\frac{4}{x}}[/latex]

26. [latex]j\left(x\right)={\left(1+x\right)}^{\frac{5}{x}}[/latex]

27. Based on the pattern you observed in the exercises above, make a conjecture as to the limit of [latex]f\left(x\right)={\left(1+x\right)}^{\frac{6}{x}}[/latex], [latex]g\left(x\right)={\left(1+x\right)}^{\frac{7}{x}}[/latex], [latex]\text{and }h\left(x\right)={\left(1+x\right)}^{\frac{n}{x}}[/latex].

For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as [latex]x[/latex] approaches [latex]a[/latex]. If the function has a limit as [latex]x[/latex] approaches [latex]a[/latex], state it. If not, discuss why there is no limit.

28. [latex]f\left(x\right)=\begin{cases}|x|-1,\hfill& \text{if }x\ne 1 \\ x^{3}, \hfill& \text{if }x=1\end{cases};\text{ }a=1[/latex]

29. [latex]f\left(x\right)=\begin{cases}\dfrac{1}{x+1},\hfill& \text{if }x=−2 \\ \left(x+1\right)^{2},\hfill& \text{if }x\ne−2\end{cases};\text{ }a=−2[/latex]

For the following exercises, use numerical evidence to determine whether the limit exists at [latex]x=a[/latex]. If not, describe the behavior of the graph of the function near [latex]x=a[/latex]. Round answers to two decimal places.

30. [latex]f\left(x\right)=\frac{{x}^{2}-4x}{16-{x}^{2}};a=4[/latex]

31. [latex]f\left(x\right)=\frac{{x}^{2}-x - 6}{{x}^{2}-9};a=3[/latex]

32. [latex]f\left(x\right)=\frac{{x}^{2}-6x - 7}{{x}^{2}- 7x};a=7[/latex]

33. [latex]f\left(x\right)=\frac{{x}^{2}-1}{{x}^{2}-3x+2};a=1[/latex]

34. [latex]f\left(x\right)=\frac{1-{x}^{2}}{{x}^{2}-3x+2};a=1[/latex]

35. [latex]f\left(x\right)=\frac{10 - 10{x}^{2}}{{x}^{2}-3x+2};a=1[/latex]

36. [latex]f\left(x\right)=\frac{x}{6{x}^{2}-5x - 6};a=\frac{3}{2}[/latex]

37. [latex]f\left(x\right)=\frac{x}{4{x}^{2}+4x+1};a=-\frac{1}{2}[/latex]

38. [latex]f\left(x\right)=\frac{2}{x - 4};\text{ }a=4[/latex]

For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as [latex]x[/latex] approaches the given value.

39. [latex]\underset{x\to 0}{\mathrm{lim}}\dfrac{7\tan x}{3x}[/latex]

40. [latex]\underset{x\to 4}{\mathrm{lim}}\dfrac{{x}^{2}}{x - 4}[/latex]

41. [latex]\underset{x\to 0}{\mathrm{lim}}\dfrac{2\sin x}{4\tan x}[/latex]

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as [latex]x[/latex] approaches [latex]a[/latex]. If the function has a limit as [latex]x[/latex] approaches [latex]a[/latex], state it. If not, discuss why there is no limit.

42. [latex]\underset{x\to 0}{\mathrm{lim}}{e}^{{e}^{\frac{1}{x}}}[/latex]

43. [latex]\underset{x\to 0}{\mathrm{lim}}{e}^{{e}^{-\frac{1}{{x}^{2}}}}[/latex]

44. [latex]\underset{x\to 0}{\mathrm{lim}}\dfrac{|x|}{x}[/latex]

45. [latex]\underset{x\to -1}{\mathrm{lim}}\dfrac{|x+1|}{x+1}[/latex]

46. [latex]\underset{x\to 5}{\mathrm{lim}}\dfrac{|x - 5|}{5-x}[/latex]

47. [latex]\underset{x\to -1}{\mathrm{lim}}\dfrac{1}{{\left(x+1\right)}^{2}}[/latex]

48. [latex]\underset{x\to 1}{\mathrm{lim}}\dfrac{1}{{\left(x - 1\right)}^{3}}[/latex]

49. [latex]\underset{x\to 0}{\mathrm{lim}}\dfrac{5}{1-{e}^{\frac{2}{x}}}[/latex]

50. Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: [latex]f\left(x\right)=\left\rvert\frac{1-x}{x}\right\rvert[/latex] and [latex]g\left(x\right)=\left\rvert\frac{1+x}{x}\right\rvert[/latex] as [latex]x[/latex] approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)[/latex] as [latex]x[/latex] approaches 0. If the functions have a limit as [latex]x[/latex] approaches 0, state it. If not, discuss why there is no limit.

51. According to the Theory of Relativity, the mass [latex]m[/latex] of a particle depends on its velocity [latex]v[/latex] . That is

[latex]m=\frac{{m}_{o}}{\sqrt{1-\left({v}^{2}/{c}^{2}\right)}}[/latex]

where [latex]{m}_{o}[/latex] is the mass when the particle is at rest and [latex]c[/latex] is the speed of light. Find the limit of the mass, [latex]m[/latex], as [latex]v[/latex] approaches [latex]{c}^{-}[/latex].

52. Allow the speed of light, [latex]c[/latex], to be equal to 1.0. If the mass, [latex]m[/latex], is 1, what occurs to [latex]m[/latex] as [latex]v\to c?[/latex] Using the values listed in the table below, make a conjecture as to what the mass is as [latex]v[/latex] approaches 1.00.