{"id":1452,"date":"2023-06-05T14:51:53","date_gmt":"2023-06-05T14:51:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-71-finding-limits-properties-of-limits\/"},"modified":"2023-06-05T14:51:53","modified_gmt":"2023-06-05T14:51:53","slug":"problem-set-71-finding-limits-properties-of-limits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/problem-set-71-finding-limits-properties-of-limits\/","title":{"raw":"Problem Set 71: Finding Limits: Properties of Limits","rendered":"Problem Set 71: Finding Limits: Properties of Limits"},"content":{"raw":"\n1. Give an example of a type of function [latex]f[\/latex] whose limit, as [latex]x[\/latex] approaches [latex]a[\/latex], is [latex]f\\left(a\\right)[\/latex].\n\n2. When direct substitution is used to evaluate the limit of a rational function as [latex]x[\/latex] approaches [latex]a[\/latex] and the result is [latex]f\\left(a\\right)=\\frac{0}{0}[\/latex], does this mean that the limit of [latex]f[\/latex] does not exist?\n\n3. What does it mean to say the limit of [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]c[\/latex], is undefined?\n\nFor the following exercises, evaluate the limits algebraically.\n\n4. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(3\\right)[\/latex]\n\n5. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{-5x}{{x}^{2}-1}\\right)[\/latex]\n\n6. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-5x+6}{x+2}\\right)[\/latex]\n\n7. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 3}\\right)[\/latex]\n\n8. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-2x - 3}{x+1}\\right)[\/latex]\n\n9. [latex]\\underset{x\\to \\frac{3}{2}}{\\mathrm{lim}}\\left(\\dfrac{6{x}^{2}-17x+12}{2x - 3}\\right)[\/latex]\n\n10. [latex]\\underset{x\\to -\\frac{7}{2}}{\\mathrm{lim}}\\left(\\dfrac{8{x}^{2}+18x - 35}{2x+7}\\right)[\/latex]\n\n11. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 5x+6}\\right)[\/latex]\n\n12. [latex]\\underset{x\\to -3}{\\mathrm{lim}}\\left(\\dfrac{-7{x}^{4}-21{x}^{3}}{-12{x}^{4}+108{x}^{2}}\\right)[\/latex]\n\n13. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}+2x - 3}{x - 3}\\right)[\/latex]\n\n14. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(3+h\\right)}^{3}-27}{h}\\right)[\/latex]\n\n15. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(2-h\\right)}^{3}-8}{h}\\right)[\/latex]\n\n16. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(h+3\\right)}^{2}-9}{h}\\right)[\/latex]\n\n17. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{5-h}-\\sqrt{5}}{h}\\right)[\/latex]\n\n18. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{3-x}-\\sqrt{3}}{x}\\right)[\/latex]\n\n19. [latex]\\underset{x\\to 9}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-81}{3-\\sqrt{x}}\\right)[\/latex]\n\n20. [latex]\\underset{x\\to 1}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{x}-{x}^{2}}{1-\\sqrt{x}}\\right)[\/latex]\n\n21. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{x}{\\sqrt{1+2x}-1}\\right)[\/latex]\n\n22. [latex]\\underset{x\\to \\frac{1}{2}}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-\\frac{1}{4}}{2x - 1}\\right)[\/latex]\n\n23. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{{x}^{3}-64}{{x}^{2}-16}\\right)[\/latex]\n\n24. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}\\left(\\dfrac{|x - 2|}{x - 2}\\right)[\/latex]\n\n25. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}\\left(\\dfrac{|x - 2|}{x - 2}\\right)[\/latex]\n\n26. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{|x - 2|}{x - 2}\\right)[\/latex]\n\n27. [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]\n\n28. [latex]\\underset{x\\to {4}^{+}}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]\n\n29. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]\n\n30. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{-8+6x-{x}^{2}}{x - 2}\\right)[\/latex]\n\nFor the following exercise, use the given information to evaluate the limits: [latex]\\underset{x\\to c}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex], [latex]\\underset{x\\to c}{\\mathrm{lim}}g\\left(x\\right)=5[\/latex]\n\n31. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\left[2f\\left(x\\right)+\\sqrt{g\\left(x\\right)}\\right][\/latex]\n\n32. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\left[3f\\left(x\\right)+\\sqrt{g\\left(x\\right)}\\right][\/latex]\n\n33. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\frac{f\\left(x\\right)}{g\\left(x\\right)}[\/latex]\n\nFor the following exercises, evaluate the following limits.\n\n34. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\cos \\left(\\pi x\\right)[\/latex]\n\n35. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\sin \\left(\\pi x\\right)[\/latex]\n\n36. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\sin \\left(\\frac{\\pi }{x}\\right)[\/latex]\n\n37. [latex]{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)[\/latex]\n\n38. [latex]{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)[\/latex]\n\n39. [latex]{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)[\/latex]\n\n40. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\dfrac{\\sqrt{x+5}-3}{x - 4}[\/latex]\n\n41. [latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\dfrac{{x}^{2}}{{x}^{2}-9}[\/latex]\n\nFor the following exercises, find the average rate of change [latex]\\frac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex].\n\n42. [latex]f\\left(x\\right)=x+1[\/latex]\n\n43. [latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]\n\n44. [latex]f\\left(x\\right)={x}^{2}+3x+4[\/latex]\n\n45. [latex]f\\left(x\\right)={x}^{2}+4x - 100[\/latex]\n\n46. [latex]f\\left(x\\right)=3{x}^{2}+1[\/latex]\n\n47. [latex]f\\left(x\\right)=\\cos \\left(x\\right)[\/latex]\n\n48. [latex]f\\left(x\\right)=2{x}^{3}-4x[\/latex]\n\n49. [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex]\n\n50. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex]\n\n51. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]\n\n52. Find an equation that could be represented by Figure 2.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 2<\/b>[\/caption]\n\n53. Find an equation that could be represented by Figure 3.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 4<\/b>[\/caption]\n\nFor the following exercises, refer to Figure 4.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 5<\/b>[\/caption]\n\n54. What is the right-hand limit of the function as [latex]x[\/latex] approaches 0?\n\n55. What is the left-hand limit of the function as [latex]x[\/latex] approaches 0?\n\n56. The position function [latex]s\\left(t\\right)=-16{t}^{2}+144t[\/latex] gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [latex]\\left[1,2\\right][\/latex] .\n\n57. The height of a projectile is given by [latex]s\\left(t\\right)=-64{t}^{2}+192t[\/latex] Find the average rate of change of the height from [latex]t=1[\/latex] second to [latex]t=1.5[\/latex] seconds.\n\n58. The amount of money in an account after [latex]t[\/latex] years compounded continuously at 4.25% interest is given by the formula [latex]A={A}_{0}{e}^{0.0425t}[\/latex], where [latex]{A}_{0}[\/latex] is the initial amount invested. Find the average rate of change of the balance of the account from [latex]t=1[\/latex] year to [latex]t=2[\/latex] years if the initial amount invested is $1,000.00.\n","rendered":"

1. Give an example of a type of function [latex]f[\/latex] whose limit, as [latex]x[\/latex] approaches [latex]a[\/latex], is [latex]f\\left(a\\right)[\/latex].<\/p>\n

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

3. What does it mean to say the limit of [latex]f\\left(x\\right)[\/latex], as [latex]x[\/latex] approaches [latex]c[\/latex], is undefined?<\/p>\n

For the following exercises, evaluate the limits algebraically.<\/p>\n

4. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(3\\right)[\/latex]<\/p>\n

5. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{-5x}{{x}^{2}-1}\\right)[\/latex]<\/p>\n

6. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-5x+6}{x+2}\\right)[\/latex]<\/p>\n

7. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 3}\\right)[\/latex]<\/p>\n

8. [latex]\\underset{x\\to -1}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-2x - 3}{x+1}\\right)[\/latex]<\/p>\n

9. [latex]\\underset{x\\to \\frac{3}{2}}{\\mathrm{lim}}\\left(\\dfrac{6{x}^{2}-17x+12}{2x - 3}\\right)[\/latex]<\/p>\n

10. [latex]\\underset{x\\to -\\frac{7}{2}}{\\mathrm{lim}}\\left(\\dfrac{8{x}^{2}+18x - 35}{2x+7}\\right)[\/latex]<\/p>\n

11. [latex]\\underset{x\\to 3}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-9}{x - 5x+6}\\right)[\/latex]<\/p>\n

12. [latex]\\underset{x\\to -3}{\\mathrm{lim}}\\left(\\dfrac{-7{x}^{4}-21{x}^{3}}{-12{x}^{4}+108{x}^{2}}\\right)[\/latex]<\/p>\n

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

14. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(3+h\\right)}^{3}-27}{h}\\right)[\/latex]<\/p>\n

15. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(2-h\\right)}^{3}-8}{h}\\right)[\/latex]<\/p>\n

16. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{{\\left(h+3\\right)}^{2}-9}{h}\\right)[\/latex]<\/p>\n

17. [latex]\\underset{h\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{5-h}-\\sqrt{5}}{h}\\right)[\/latex]<\/p>\n

18. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{\\sqrt{3-x}-\\sqrt{3}}{x}\\right)[\/latex]<\/p>\n

19. [latex]\\underset{x\\to 9}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-81}{3-\\sqrt{x}}\\right)[\/latex]<\/p>\n

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

21. [latex]\\underset{x\\to 0}{\\mathrm{lim}}\\left(\\dfrac{x}{\\sqrt{1+2x}-1}\\right)[\/latex]<\/p>\n

22. [latex]\\underset{x\\to \\frac{1}{2}}{\\mathrm{lim}}\\left(\\dfrac{{x}^{2}-\\frac{1}{4}}{2x - 1}\\right)[\/latex]<\/p>\n

23. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{{x}^{3}-64}{{x}^{2}-16}\\right)[\/latex]<\/p>\n

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

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

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

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

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

29. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\left(\\dfrac{|x - 4|}{4-x}\\right)[\/latex]<\/p>\n

30. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\left(\\dfrac{-8+6x-{x}^{2}}{x - 2}\\right)[\/latex]<\/p>\n

For the following exercise, use the given information to evaluate the limits: [latex]\\underset{x\\to c}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex], [latex]\\underset{x\\to c}{\\mathrm{lim}}g\\left(x\\right)=5[\/latex]<\/p>\n

31. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\left[2f\\left(x\\right)+\\sqrt{g\\left(x\\right)}\\right][\/latex]<\/p>\n

32. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\left[3f\\left(x\\right)+\\sqrt{g\\left(x\\right)}\\right][\/latex]<\/p>\n

33. [latex]\\underset{x\\to c}{\\mathrm{lim}}\\frac{f\\left(x\\right)}{g\\left(x\\right)}[\/latex]<\/p>\n

For the following exercises, evaluate the following limits.<\/p>\n

34. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\cos \\left(\\pi x\\right)[\/latex]<\/p>\n

35. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\sin \\left(\\pi x\\right)[\/latex]<\/p>\n

36. [latex]\\underset{x\\to 2}{\\mathrm{lim}}\\sin \\left(\\frac{\\pi }{x}\\right)[\/latex]<\/p>\n

37. [latex]{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)[\/latex]<\/p>\n

38. [latex]{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)[\/latex]<\/p>\n

39. [latex]{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)[\/latex]<\/p>\n

40. [latex]\\underset{x\\to 4}{\\mathrm{lim}}\\dfrac{\\sqrt{x+5}-3}{x - 4}[\/latex]<\/p>\n

41. [latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\dfrac{{x}^{2}}{{x}^{2}-9}[\/latex]<\/p>\n

For the following exercises, find the average rate of change [latex]\\frac{f\\left(x+h\\right)-f\\left(x\\right)}{h}[\/latex].<\/p>\n

42. [latex]f\\left(x\\right)=x+1[\/latex]<\/p>\n

43. [latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\n

44. [latex]f\\left(x\\right)={x}^{2}+3x+4[\/latex]<\/p>\n

45. [latex]f\\left(x\\right)={x}^{2}+4x - 100[\/latex]<\/p>\n

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

47. [latex]f\\left(x\\right)=\\cos \\left(x\\right)[\/latex]<\/p>\n

48. [latex]f\\left(x\\right)=2{x}^{3}-4x[\/latex]<\/p>\n

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

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

51. [latex]f\\left(x\\right)=\\sqrt{x}[\/latex]<\/p>\n

52. Find an equation that could be represented by Figure 2.<\/p>\n

\"Graph<\/p>\n

Figure 2<\/b><\/p>\n<\/div>\n

53. Find an equation that could be represented by Figure 3.<\/p>\n

\"Graph<\/p>\n

Figure 4<\/b><\/p>\n<\/div>\n

For the following exercises, refer to Figure 4.<\/p>\n

\"Graph<\/p>\n

Figure 5<\/b><\/p>\n<\/div>\n

54. What is the right-hand limit of the function as [latex]x[\/latex] approaches 0?<\/p>\n

55. What is the left-hand limit of the function as [latex]x[\/latex] approaches 0?<\/p>\n

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

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

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

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