{"id":15788,"date":"2019-09-05T19:51:44","date_gmt":"2019-09-05T19:51:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/?post_type=chapter&p=15788"},"modified":"2025-02-05T05:25:27","modified_gmt":"2025-02-05T05:25:27","slug":"problem-set-72-continuity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/problem-set-72-continuity\/","title":{"raw":"Problem Set 72: Continuity","rendered":"Problem Set 72: Continuity"},"content":{"raw":"1. State in your own words what it means for a function [latex]f[\/latex] to be continuous at [latex]x=c[\/latex].\r\n\r\n2.\u00a0State in your own words what it means for a function to be continuous on the interval [latex]\\left(a,b\\right)[\/latex].\r\n\r\nFor the following exercises, determine why the function [latex]f[\/latex] is discontinuous at a given point [latex]a[\/latex] on the graph. State which condition fails.\r\n\r\n3. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }|\\text{ }x+3\\text{ }|,a=-3[\/latex]\r\n\r\n4.\u00a0[latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }|\\text{ }5x - 2\\text{ }|,a=\\frac{2}{5}[\/latex]\r\n\r\n5. [latex]f\\left(x\\right)=\\frac{{x}^{2}-16}{x+4},a=-4[\/latex]\r\n\r\n6.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{2}-16x}{x},a=0[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)=\\begin{cases}x,\\hfill& x\\neq 3 \\\\ 2x, \\hfill& x=3\\end{cases}a=3[\/latex]\r\n\r\n8.\u00a0[latex]f\\left(x\\right)=\\begin{cases}5, \\hfill& x\\neq 0 \\\\ 3, \\hfill& x=0\\end{cases} a=0[\/latex]\r\n\r\n9.\u00a0[latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{2-x}, \\hfill& x\\neq 2 \\\\ 3, \\hfill& x=2\\end{cases}a=2[\/latex]\r\n\r\n10. [latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{x+6}, \\hfill& x=-6 \\\\ x^{2}, \\hfill& x\\neq -6\\end{cases}a=-6[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)=\\begin{cases}3+x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]\r\n\r\n12.\u00a0[latex]f\\left(x\\right)=\\begin{cases}3-x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ 2x^{2}, \\hfill& x>1\\end{cases} a=1[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=\\begin{cases}3+2x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ -x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]\r\n\r\n14.\u00a0[latex]f\\left(x\\right)=\\begin{cases}x^{2}, \\hfill& x<-2 \\\\ 2x+1, \\hfill& x=-2 \\\\ x^{3}, \\hfill& x>-2\\end{cases}a=-2[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=\\begin{cases}\\frac{x^{2}-9}{x+3}, \\hfill& x<-3 \\\\ x-9, \\hfill& x=-3 \\\\ \\frac{1}{x}, \\hfill& x>-3\\end{cases}a=-3[\/latex]\r\n\r\n16. [latex]f\\left(x\\right)=\\begin{cases}\\frac{x^{2}-9}{x+3}, \\hfill& x<-3 \\\\ x-9, \\hfill& x=-3 \\\\ -6, \\hfill& x>-3\\end{cases}a=3[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=\\frac{{x}^{2}-4}{x - 2},\\text{ }a=2[\/latex]\r\n\r\n18.\u00a0[latex]f\\left(x\\right)=\\frac{25-{x}^{2}}{{x}^{2}-10x+25},\\text{ }a=5[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=\\frac{{x}^{3}-9x}{{x}^{2}+11x+24},\\text{ }a=-3[\/latex]\r\n\r\n20.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{3}-27}{{x}^{2}-3x},\\text{ }a=3[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)=\\frac{x}{|x|},\\text{ }a=0[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)=\\frac{2|x+2|}{x+2},\\text{ }a=-2[\/latex]\r\n\r\nFor the following exercises, determine whether or not the given function [latex]f[\/latex] is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.\r\n\r\n23. [latex]f\\left(x\\right)={x}^{3}-2x - 15[\/latex]\r\n\r\n24.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{2}-2x - 15}{x - 5}[\/latex]\r\n\r\n25. [latex]f\\left(x\\right)=2\\cdot {3}^{x+4}[\/latex]\r\n\r\n26.\u00a0[latex]f\\left(x\\right)=\\mathrm{-sin}\\left(3x\\right)[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)=\\frac{|x - 2|}{{x}^{2}-2x}[\/latex]\r\n\r\n28.\u00a0[latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex]\r\n\r\n29. [latex]f\\left(x\\right)=2x+\\frac{5}{x}[\/latex]\r\n\r\n30.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]\r\n\r\n31. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }{x}^{2}[\/latex]\r\n\r\n32.\u00a0[latex]f\\left(x\\right)={e}^{2x}[\/latex]\r\n\r\n33. [latex]f\\left(x\\right)=\\sqrt{x - 4}[\/latex]\r\n\r\n34.\u00a0[latex]f\\left(x\\right)=\\sec \\left(x\\right)-3[\/latex] .\r\n\r\n35. [latex]f\\left(x\\right)={x}^{2}+\\sin \\left(x\\right)[\/latex]\r\n\r\n36.\u00a0Determine the values of [latex]b[\/latex] and [latex]c[\/latex] such that the following function is continuous on the entire real number line.\r\n

[latex]f\\left(x\\right)=\\begin{cases}x+1, \\hfill& {1 }<{x }<{3 }\\\\ x^{2}+bx+c, \\hfill& |x-2|\\geq 1\\end{cases}[\/latex]<\/p>\r\nFor the following exercises, refer to Figure 15. Each square represents one square unit. For each value of [latex]a[\/latex], determine which of the three conditions of continuity are satisfied at [latex]x=a[\/latex] and which are not.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 15<\/b>[\/caption]\r\n\r\n37. [latex]x=-3[\/latex]\r\n\r\n38.\u00a0[latex]x=2[\/latex]\r\n\r\n39. [latex]x=4[\/latex]\r\n\r\nFor the following exercises, use a graphing utility to graph the function [latex]f\\left(x\\right)=\\sin \\left(\\frac{12\\pi }{x}\\right)[\/latex] as in Figure 16. Set the x<\/em>-axis a short distance before and after 0 to illustrate the point of discontinuity.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 16<\/b>[\/caption]40. Which conditions for continuity fail at the point of discontinuity?\r\n\r\n41. Evaluate [latex]f\\left(0\\right)[\/latex].\r\n\r\n42.\u00a0Solve for [latex]x[\/latex] if [latex]f\\left(x\\right)=0[\/latex].\r\n\r\n43. What is the domain of [latex]f\\left(x\\right)?[\/latex]\r\n\r\nFor the following exercises, consider the function shown in Figure 17.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]\"Graph Figure 17<\/b>[\/caption]\r\n\r\n44. At what x<\/em>-coordinates is the function discontinuous?\r\n\r\n45. What condition of continuity is violated at these points?\r\n\r\n46.\u00a0Consider the function shown in Figure 18. At what x<\/em>-coordinates is the function discontinuous? What condition(s) of continuity were violated?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 18<\/b>[\/caption]\r\n\r\n47. Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at [latex]x=-7[\/latex] and [latex]x=1[\/latex].\r\n\r\n48.\u00a0The function [latex]f\\left(x\\right)=\\frac{{x}^{3}-1}{x - 1}[\/latex] is graphed in Figure 19. It appears to be continuous on the interval [latex]\\left[-3,3\\right][\/latex], but there is an x<\/em>-value on that interval at which the function is discontinuous. Determine the value of [latex]x[\/latex] at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]\"Graph Figure 19<\/b>[\/caption]\r\n\r\n49. Find the limit [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex] and determine if the following function is continuous at [latex]x=1:[\/latex]\r\n\r\n50.\u00a0The function is discontinuous at [latex]x=1[\/latex] because the limit as [latex]x[\/latex] approaches 1 is 5 and [latex]f\\left(1\\right)=2[\/latex].\r\n\r\n51. The graph of [latex]f\\left(x\\right)=\\frac{\\sin \\left(2x\\right)}{x}[\/latex] is shown in Figure 20. Is the function [latex]f\\left(x\\right)[\/latex] continuous at [latex]x=0?[\/latex] Why or why not?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"Graph Figure 20<\/b>[\/caption]","rendered":"

1. State in your own words what it means for a function [latex]f[\/latex] to be continuous at [latex]x=c[\/latex].<\/p>\n

2.\u00a0State in your own words what it means for a function to be continuous on the interval [latex]\\left(a,b\\right)[\/latex].<\/p>\n

For the following exercises, determine why the function [latex]f[\/latex] is discontinuous at a given point [latex]a[\/latex] on the graph. State which condition fails.<\/p>\n

3. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }|\\text{ }x+3\\text{ }|,a=-3[\/latex]<\/p>\n

4.\u00a0[latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }|\\text{ }5x - 2\\text{ }|,a=\\frac{2}{5}[\/latex]<\/p>\n

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

6.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{2}-16x}{x},a=0[\/latex]<\/p>\n

7. [latex]f\\left(x\\right)=\\begin{cases}x,\\hfill& x\\neq 3 \\\\ 2x, \\hfill& x=3\\end{cases}a=3[\/latex]<\/p>\n

8.\u00a0[latex]f\\left(x\\right)=\\begin{cases}5, \\hfill& x\\neq 0 \\\\ 3, \\hfill& x=0\\end{cases} a=0[\/latex]<\/p>\n

9.\u00a0[latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{2-x}, \\hfill& x\\neq 2 \\\\ 3, \\hfill& x=2\\end{cases}a=2[\/latex]<\/p>\n

10. [latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{x+6}, \\hfill& x=-6 \\\\ x^{2}, \\hfill& x\\neq -6\\end{cases}a=-6[\/latex]<\/p>\n

11. [latex]f\\left(x\\right)=\\begin{cases}3+x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]<\/p>\n

12.\u00a0[latex]f\\left(x\\right)=\\begin{cases}3-x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ 2x^{2}, \\hfill& x>1\\end{cases} a=1[\/latex]<\/p>\n

13. [latex]f\\left(x\\right)=\\begin{cases}3+2x, \\hfill& x<1 \\\\ x, \\hfill& x=1 \\\\ -x^{2}, \\hfill& x>1\\end{cases}a=1[\/latex]<\/p>\n

14.\u00a0[latex]f\\left(x\\right)=\\begin{cases}x^{2}, \\hfill& x<-2 \\\\ 2x+1, \\hfill& x=-2 \\\\ x^{3}, \\hfill& x>-2\\end{cases}a=-2[\/latex]<\/p>\n

15. [latex]f\\left(x\\right)=\\begin{cases}\\frac{x^{2}-9}{x+3}, \\hfill& x<-3 \\\\ x-9, \\hfill& x=-3 \\\\ \\frac{1}{x}, \\hfill& x>-3\\end{cases}a=-3[\/latex]<\/p>\n

16. [latex]f\\left(x\\right)=\\begin{cases}\\frac{x^{2}-9}{x+3}, \\hfill& x<-3 \\\\ x-9, \\hfill& x=-3 \\\\ -6, \\hfill& x>-3\\end{cases}a=3[\/latex]<\/p>\n

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

18.\u00a0[latex]f\\left(x\\right)=\\frac{25-{x}^{2}}{{x}^{2}-10x+25},\\text{ }a=5[\/latex]<\/p>\n

19. [latex]f\\left(x\\right)=\\frac{{x}^{3}-9x}{{x}^{2}+11x+24},\\text{ }a=-3[\/latex]<\/p>\n

20.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{3}-27}{{x}^{2}-3x},\\text{ }a=3[\/latex]<\/p>\n

21. [latex]f\\left(x\\right)=\\frac{x}{|x|},\\text{ }a=0[\/latex]<\/p>\n

22.\u00a0[latex]f\\left(x\\right)=\\frac{2|x+2|}{x+2},\\text{ }a=-2[\/latex]<\/p>\n

For the following exercises, determine whether or not the given function [latex]f[\/latex] is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.<\/p>\n

23. [latex]f\\left(x\\right)={x}^{3}-2x - 15[\/latex]<\/p>\n

24.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{2}-2x - 15}{x - 5}[\/latex]<\/p>\n

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

26.\u00a0[latex]f\\left(x\\right)=\\mathrm{-sin}\\left(3x\\right)[\/latex]<\/p>\n

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

28.\u00a0[latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex]<\/p>\n

29. [latex]f\\left(x\\right)=2x+\\frac{5}{x}[\/latex]<\/p>\n

30.\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/p>\n

31. [latex]f\\left(x\\right)=\\mathrm{ln}\\text{ }{x}^{2}[\/latex]<\/p>\n

32.\u00a0[latex]f\\left(x\\right)={e}^{2x}[\/latex]<\/p>\n

33. [latex]f\\left(x\\right)=\\sqrt{x - 4}[\/latex]<\/p>\n

34.\u00a0[latex]f\\left(x\\right)=\\sec \\left(x\\right)-3[\/latex] .<\/p>\n

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

36.\u00a0Determine the values of [latex]b[\/latex] and [latex]c[\/latex] such that the following function is continuous on the entire real number line.<\/p>\n

[latex]f\\left(x\\right)=\\begin{cases}x+1, \\hfill& {1 }<{x }<{3 }\\\\ x^{2}+bx+c, \\hfill& |x-2|\\geq 1\\end{cases}[\/latex]<\/p>\n

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of [latex]a[\/latex], determine which of the three conditions of continuity are satisfied at [latex]x=a[\/latex] and which are not.<\/p>\n

\"Graph<\/p>\n

Figure 15<\/b><\/p>\n<\/div>\n

37. [latex]x=-3[\/latex]<\/p>\n

38.\u00a0[latex]x=2[\/latex]<\/p>\n

39. [latex]x=4[\/latex]<\/p>\n

For the following exercises, use a graphing utility to graph the function [latex]f\\left(x\\right)=\\sin \\left(\\frac{12\\pi }{x}\\right)[\/latex] as in Figure 16. Set the x<\/em>-axis a short distance before and after 0 to illustrate the point of discontinuity.<\/p>\n

\"Graph<\/p>\n

Figure 16<\/b><\/p>\n<\/div>\n

40. Which conditions for continuity fail at the point of discontinuity?<\/p>\n

41. Evaluate [latex]f\\left(0\\right)[\/latex].<\/p>\n

42.\u00a0Solve for [latex]x[\/latex] if [latex]f\\left(x\\right)=0[\/latex].<\/p>\n

43. What is the domain of [latex]f\\left(x\\right)?[\/latex]<\/p>\n

For the following exercises, consider the function shown in Figure 17.<\/p>\n

\"Graph<\/p>\n

Figure 17<\/b><\/p>\n<\/div>\n

44. At what x<\/em>-coordinates is the function discontinuous?<\/p>\n

45. What condition of continuity is violated at these points?<\/p>\n

46.\u00a0Consider the function shown in Figure 18. At what x<\/em>-coordinates is the function discontinuous? What condition(s) of continuity were violated?<\/p>\n

\"Graph<\/p>\n

Figure 18<\/b><\/p>\n<\/div>\n

47. Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at [latex]x=-7[\/latex] and [latex]x=1[\/latex].<\/p>\n

48.\u00a0The function [latex]f\\left(x\\right)=\\frac{{x}^{3}-1}{x - 1}[\/latex] is graphed in Figure 19. It appears to be continuous on the interval [latex]\\left[-3,3\\right][\/latex], but there is an x<\/em>-value on that interval at which the function is discontinuous. Determine the value of [latex]x[\/latex] at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.<\/p>\n

\"Graph<\/p>\n

Figure 19<\/b><\/p>\n<\/div>\n

49. Find the limit [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex] and determine if the following function is continuous at [latex]x=1:[\/latex]<\/p>\n

50.\u00a0The function is discontinuous at [latex]x=1[\/latex] because the limit as [latex]x[\/latex] approaches 1 is 5 and [latex]f\\left(1\\right)=2[\/latex].<\/p>\n

51. The graph of [latex]f\\left(x\\right)=\\frac{\\sin \\left(2x\\right)}{x}[\/latex] is shown in Figure 20. Is the function [latex]f\\left(x\\right)[\/latex] continuous at [latex]x=0?[\/latex] Why or why not?<\/p>\n

\"Graph<\/p>\n

Figure 20<\/b><\/p>\n<\/div>\n\n\t\t\t

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