{"id":1450,"date":"2023-06-05T14:51:52","date_gmt":"2023-06-05T14:51:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-70-finding-limits-numerical-and-graphical-approaches\/"},"modified":"2023-06-05T14:51:52","modified_gmt":"2023-06-05T14:51:52","slug":"problem-set-70-finding-limits-numerical-and-graphical-approaches","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-70-finding-limits-numerical-and-graphical-approaches\/","title":{"raw":"Problem Set 70: Finding Limits: Numerical and Graphical Approaches","rendered":"Problem Set 70: Finding Limits: Numerical and Graphical Approaches"},"content":{"raw":"\n1. Explain the difference between a value at [latex]x=a[\/latex] and the limit as [latex]x[\/latex] approaches [latex]a[\/latex].\n\n2. 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.\n\nFor the following exercises, estimate the functional values and the limits from the graph of the function [latex]f[\/latex] provided in Figure 14.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"A Figure 14<\/b>[\/caption]\n\n3. [latex]\\underset{x\\to -{2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n4. [latex]\\underset{x\\to -{2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n5. [latex]\\underset{x\\to -2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n6. [latex]f\\left(-2\\right)[\/latex]\n\n7. [latex]\\underset{x\\to -{1}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n8. [latex]\\underset{x\\to {1}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n9. [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n10. [latex]f\\left(1\\right)[\/latex]\n\n11. [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n12. [latex]\\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n13. [latex]\\underset{x\\to 4}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n\n14. [latex]f\\left(4\\right)[\/latex]\n\nFor the following exercises, draw the graph of a function from the functional values and limits provided.\n\n15. [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].\n\n16. [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]\n\n17. [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]\n\n18. [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].\n\n19. [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]\n\n20. [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]\n\n21. [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].\n\nFor the following exercises, use a graphing calculator to determine the limit to 5 decimal places as [latex]x[\/latex] approaches 0.\n\n22. [latex]f\\left(x\\right)={\\left(1+x\\right)}^{\\frac{1}{x}}[\/latex]\n\n23. [latex]g\\left(x\\right)={\\left(1+x\\right)}^{\\frac{2}{x}}[\/latex]\n\n24. [latex]h\\left(x\\right)={\\left(1+x\\right)}^{\\frac{3}{x}}[\/latex]\n\n25. [latex]i\\left(x\\right)={\\left(1+x\\right)}^{\\frac{4}{x}}[\/latex]\n\n26. [latex]j\\left(x\\right)={\\left(1+x\\right)}^{\\frac{5}{x}}[\/latex]\n\n27. 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].\n\nFor 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.\n\n28. [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]\n\n29. [latex]f\\left(x\\right)=\\begin{cases}\\dfrac{1}{x+1},\\hfill& \\text{if }x=\u22122 \\\\ \\left(x+1\\right)^{2},\\hfill& \\text{if }x\\ne\u22122\\end{cases};\\text{ }a=\u22122[\/latex]\n\nFor 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.\n\n30. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4x}{16-{x}^{2}};a=4[\/latex]\n\n31. [latex]f\\left(x\\right)=\\frac{{x}^{2}-x - 6}{{x}^{2}-9};a=3[\/latex]\n\n32. [latex]f\\left(x\\right)=\\frac{{x}^{2}-6x - 7}{{x}^{2}- 7x};a=7[\/latex]\n\n33. [latex]f\\left(x\\right)=\\frac{{x}^{2}-1}{{x}^{2}-3x+2};a=1[\/latex]\n\n34. [latex]f\\left(x\\right)=\\frac{1-{x}^{2}}{{x}^{2}-3x+2};a=1[\/latex]\n\n35. [latex]f\\left(x\\right)=\\frac{10 - 10{x}^{2}}{{x}^{2}-3x+2};a=1[\/latex]\n\n36. [latex]f\\left(x\\right)=\\frac{x}{6{x}^{2}-5x - 6};a=\\frac{3}{2}[\/latex]\n\n37. [latex]f\\left(x\\right)=\\frac{x}{4{x}^{2}+4x+1};a=-\\frac{1}{2}[\/latex]\n\n38. [latex]f\\left(x\\right)=\\frac{2}{x - 4};\\text{ }a=4[\/latex]\n\nFor 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.\n\n39. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{7\\tan x}{3x}[\/latex]\n\n40. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\dfrac{{x}^{2}}{x - 4}[\/latex]\n\n41. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{2\\sin x}{4\\tan x}[\/latex]\n\nFor 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.\n\n42. [latex]\\underset{x\\to 0}{\\mathrm{lim}}{e}^{{e}^{\\frac{1}{x}}}[\/latex]\n\n43. [latex]\\underset{x\\to 0}{\\mathrm{lim}}{e}^{{e}^{-\\frac{1}{{x}^{2}}}}[\/latex]\n\n44. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{|x|}{x}[\/latex]\n\n45. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\dfrac{|x+1|}{x+1}[\/latex]\n\n46. [latex]\\underset{x\\to 5}{\\mathrm{lim}}\\dfrac{|x - 5|}{5-x}[\/latex]\n\n47. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\dfrac{1}{{\\left(x+1\\right)}^{2}}[\/latex]\n\n48. [latex]\\underset{x\\to 1}{\\mathrm{lim}}\\dfrac{1}{{\\left(x - 1\\right)}^{3}}[\/latex]\n\n49. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\dfrac{5}{1-{e}^{\\frac{2}{x}}}[\/latex]\n\n50. 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.\n\n51. According to the Theory of Relativity, the mass [latex]m[\/latex] of a particle depends on its velocity [latex]v[\/latex] . That is\n

[latex]m=\\frac{{m}_{o}}{\\sqrt{1-\\left({v}^{2}\/{c}^{2}\\right)}}[\/latex]<\/p>\nwhere [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].\n\n52. 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.\n\n\n\n\n\n\n\n\n\n
[latex]v[\/latex] <\/strong><\/td>\n [latex]m[\/latex] <\/strong><\/td>\n<\/tr>\n
0.5<\/td>\n1.15<\/td>\n<\/tr>\n
0.9<\/td>\n2.29<\/td>\n<\/tr>\n
0.95<\/td>\n3.20<\/td>\n<\/tr>\n
0.99<\/td>\n7.09<\/td>\n<\/tr>\n
0.999<\/td>\n22.36<\/td>\n<\/tr>\n
0.99999<\/td>\n223.61<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","rendered":"

1. Explain the difference between a value at [latex]x=a[\/latex] and the limit as [latex]x[\/latex] approaches [latex]a[\/latex].<\/p>\n

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.<\/p>\n

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

\"A<\/p>\n

Figure 14<\/b><\/p>\n<\/div>\n

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

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

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

6. [latex]f\\left(-2\\right)[\/latex]<\/p>\n

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

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

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

10. [latex]f\\left(1\\right)[\/latex]<\/p>\n

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

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

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

14. [latex]f\\left(4\\right)[\/latex]<\/p>\n

For the following exercises, draw the graph of a function from the functional values and limits provided.<\/p>\n

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].<\/p>\n

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]<\/p>\n

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]<\/p>\n

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].<\/p>\n

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]<\/p>\n

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]<\/p>\n

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].<\/p>\n

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

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

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

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

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

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

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].<\/p>\n

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.<\/p>\n

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]<\/p>\n

29. [latex]f\\left(x\\right)=\\begin{cases}\\dfrac{1}{x+1},\\hfill& \\text{if }x=\u22122 \\\\ \\left(x+1\\right)^{2},\\hfill& \\text{if }x\\ne\u22122\\end{cases};\\text{ }a=\u22122[\/latex]<\/p>\n

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.<\/p>\n

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

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

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

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

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

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

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

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

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

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.<\/p>\n

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

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

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

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.<\/p>\n

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

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

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

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

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

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

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

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

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.<\/p>\n

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

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

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].<\/p>\n

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.<\/p>\n\n\n\n\n\n\n\n\n\n
[latex]v[\/latex] <\/strong><\/td>\n [latex]m[\/latex] <\/strong><\/td>\n<\/tr>\n
0.5<\/td>\n1.15<\/td>\n<\/tr>\n
0.9<\/td>\n2.29<\/td>\n<\/tr>\n
0.95<\/td>\n3.20<\/td>\n<\/tr>\n
0.99<\/td>\n7.09<\/td>\n<\/tr>\n
0.999<\/td>\n22.36<\/td>\n<\/tr>\n
0.99999<\/td>\n223.61<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\t\t\t
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