{"id":2487,"date":"2018-02-01T15:33:28","date_gmt":"2018-02-01T15:33:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/?post_type=chapter&p=2487"},"modified":"2018-04-25T16:02:28","modified_gmt":"2018-04-25T16:02:28","slug":"chapter-2-review-exercises","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/chapter\/chapter-2-review-exercises\/","title":{"raw":"Chapter 2 Review Exercises","rendered":"Chapter 2 Review Exercises"},"content":{"raw":"\r\n

True or False<\/em>. In the following exercises, justify your answer with a proof or a counterexample.<\/p>\r\n\r\n

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1.\u00a0<\/strong>A function has to be continuous at [latex]x=a[\/latex] if the [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>You can use the quotient rule to evaluate [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1170571653983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571653983\"]\r\n

False<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>If there is a vertical asymptote at [latex]x=a[\/latex] for the function [latex]f(x)[\/latex], then [latex]f[\/latex] is undefined at the point [latex]x=a[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist, then [latex]f[\/latex] is undefined at the point [latex]x=a[\/latex].<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1170571597426\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571597426\"]\r\n

False. A removable discontinuity is possible.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>Using the graph, find each limit or explain why the limit does not exist.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]\\underset{x\\to -1}{\\lim}f(x)[\/latex]<\/li>\r\n \t
  2. [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]<\/li>\r\n \t
  3. [latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/li>\r\n \t
  4. [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/li>\r\n<\/ol>\r\n\"A<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

    In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.<\/p>\r\n\r\n

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    6.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\frac{2x^2-3x-2}{x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572330176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572330176\"]\r\n

    5<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    7.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}3x^2-2x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    8.\u00a0<\/strong>[latex]\\underset{x\\to 3}{\\lim}\\frac{x^3-2x^2-1}{3x-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170571733970\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571733970\"]\r\n

    [latex]8\/7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    9.\u00a0<\/strong>[latex]\\underset{x\\to \\pi\/2}{\\lim}\\frac{\\cot x}{\\cos x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    10.\u00a0<\/strong>[latex]\\underset{x\\to -5}{\\lim}\\frac{x^2+25}{x+5}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170571636177\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571636177\"]\r\n

    DNE<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    11.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\frac{3x^2-2x-8}{x^2-4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    12.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2-1}{x^3-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572510140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572510140\"]\r\n

    [latex]2\/3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    13.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2-1}{\\sqrt{x}-1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    14.\u00a0<\/strong>[latex]\\underset{x\\to 4}{\\lim}\\frac{4-x}{\\sqrt{x}-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572419312\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572419312\"]\r\n

    \u22124<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    15.\u00a0<\/strong>[latex]\\underset{x\\to 4}{\\lim}\\frac{1}{\\sqrt{x}-2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    In the following exercises, use the squeeze theorem to prove the limit.<\/p>\r\n\r\n

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    16.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^2\\cos(2\\pi x)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572333059\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572333059\"]\r\n

    Since [latex]-1\\le \\cos (2\\pi x)\\le 1[\/latex], then [latex]-x^2\\le x^2\\cos(2\\pi x)\\le x^2[\/latex]. Since [latex]\\underset{x\\to 0}{\\lim}x^2=0=\\underset{x\\to 0}{\\lim}-x^2[\/latex], it follows that [latex]\\underset{x\\to 0}{\\lim}x^2\\cos(2\\pi x)=0[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    17.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^3\\sin(\\frac{\\pi}{x})=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    18.\u00a0<\/strong>Determine the domain such that the function [latex]f(x)=\\sqrt{x-2}+xe^x[\/latex] is continuous over its domain.<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170571697178\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571697178\"]\r\n

    [latex][2,\\infty)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    In the following exercises, determine the value of [latex]c[\/latex] such that the function remains continuous. Draw your resulting function to ensure it is continuous.<\/p>\r\n\r\n

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    19.\u00a0<\/strong>[latex]f(x)=\\begin{cases} x^2+1 & \\text{if} \\, x>c \\\\ 2x & \\text{if} \\, x \\le c \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    20.\u00a0<\/strong>[latex]f(x)=\\begin{cases} \\sqrt{x+1} & \\text{if} \\, x > -1 \\\\ x^2+c & \\text{if} \\, x \\le -1 \\end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572609485\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572609485\"]\r\n

    [latex]c=-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

    In the following exercises, use the precise definition of limit to prove the limit.<\/p>\r\n\r\n

    \r\n
    \r\n

    21.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}(8x+16)=24[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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    22.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^3=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170571710699\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571710699\"]\r\n

    [latex]\\delta =\\sqrt[3]{\\epsilon}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    23.\u00a0<\/strong>A ball is thrown into the air and the vertical position is given by [latex]x(t)=-4.9t^2+25t+5[\/latex]. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    24.\u00a0<\/strong>A particle moving along a line has a displacement according to the function [latex]x(t)=t^2-2t+4[\/latex], where [latex]x[\/latex] is measured in meters and [latex]t[\/latex] is measured in seconds. Find the average velocity over the time period [latex]t=[0,2][\/latex].<\/p>\r\n\r\n<\/div>\r\n

    [reveal-answer q=\"fs-id1170572386121\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572386121\"]\r\n

    [latex]0[\/latex] m\/sec<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    25.\u00a0<\/strong>From the previous exercises, estimate the instantaneous velocity at [latex]t=2[\/latex] by checking the average velocity within [latex]t=0.01[\/latex] sec.<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

    True or False<\/em>. In the following exercises, justify your answer with a proof or a counterexample.<\/p>\n

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    \n

    1.\u00a0<\/strong>A function has to be continuous at [latex]x=a[\/latex] if the [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] exists.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    2.\u00a0<\/strong>You can use the quotient rule to evaluate [latex]\\underset{x\\to 0}{\\lim}\\frac{\\sin x}{x}[\/latex].<\/p>\n<\/div>\n

    \n
    Show Solution<\/span><\/p>\n
    \n

    False<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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    3.\u00a0<\/strong>If there is a vertical asymptote at [latex]x=a[\/latex] for the function [latex]f(x)[\/latex], then [latex]f[\/latex] is undefined at the point [latex]x=a[\/latex].<\/p>\n<\/div>\n<\/div>\n

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    4.\u00a0<\/strong>If [latex]\\underset{x\\to a}{\\lim}f(x)[\/latex] does not exist, then [latex]f[\/latex] is undefined at the point [latex]x=a[\/latex].<\/p>\n<\/div>\n

    \n
    Show Solution<\/span><\/p>\n
    \n

    False. A removable discontinuity is possible.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

    \n
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    5.\u00a0<\/strong>Using the graph, find each limit or explain why the limit does not exist.<\/p>\n

      \n
    1. [latex]\\underset{x\\to -1}{\\lim}f(x)[\/latex]<\/li>\n
    2. [latex]\\underset{x\\to 1}{\\lim}f(x)[\/latex]<\/li>\n
    3. [latex]\\underset{x\\to 0^+}{\\lim}f(x)[\/latex]<\/li>\n
    4. [latex]\\underset{x\\to 2}{\\lim}f(x)[\/latex]<\/li>\n<\/ol>\n

      \"A<\/span><\/p>\n<\/div>\n<\/div>\n

      In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.<\/p>\n

      \n
      \n

      6.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\frac{2x^2-3x-2}{x-2}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

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

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      7.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}3x^2-2x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
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      8.\u00a0<\/strong>[latex]\\underset{x\\to 3}{\\lim}\\frac{x^3-2x^2-1}{3x-2}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex]8\/7[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      9.\u00a0<\/strong>[latex]\\underset{x\\to \\pi\/2}{\\lim}\\frac{\\cot x}{\\cos x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
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      10.\u00a0<\/strong>[latex]\\underset{x\\to -5}{\\lim}\\frac{x^2+25}{x+5}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      DNE<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      11.\u00a0<\/strong>[latex]\\underset{x\\to 2}{\\lim}\\frac{3x^2-2x-8}{x^2-4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
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      12.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2-1}{x^3-1}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex]2\/3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      13.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}\\frac{x^2-1}{\\sqrt{x}-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      14.\u00a0<\/strong>[latex]\\underset{x\\to 4}{\\lim}\\frac{4-x}{\\sqrt{x}-2}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      \u22124<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      15.\u00a0<\/strong>[latex]\\underset{x\\to 4}{\\lim}\\frac{1}{\\sqrt{x}-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      In the following exercises, use the squeeze theorem to prove the limit.<\/p>\n

      \n
      \n

      16.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^2\\cos(2\\pi x)=0[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      Since [latex]-1\\le \\cos (2\\pi x)\\le 1[\/latex], then [latex]-x^2\\le x^2\\cos(2\\pi x)\\le x^2[\/latex]. Since [latex]\\underset{x\\to 0}{\\lim}x^2=0=\\underset{x\\to 0}{\\lim}-x^2[\/latex], it follows that [latex]\\underset{x\\to 0}{\\lim}x^2\\cos(2\\pi x)=0[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      17.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^3\\sin(\\frac{\\pi}{x})=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      18.\u00a0<\/strong>Determine the domain such that the function [latex]f(x)=\\sqrt{x-2}+xe^x[\/latex] is continuous over its domain.<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex][2,\\infty)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      In the following exercises, determine the value of [latex]c[\/latex] such that the function remains continuous. Draw your resulting function to ensure it is continuous.<\/p>\n

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      19.\u00a0<\/strong>[latex]f(x)=\\begin{cases} x^2+1 & \\text{if} \\, x>c \\\\ 2x & \\text{if} \\, x \\le c \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      20.\u00a0<\/strong>[latex]f(x)=\\begin{cases} \\sqrt{x+1} & \\text{if} \\, x > -1 \\\\ x^2+c & \\text{if} \\, x \\le -1 \\end{cases}[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex]c=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      In the following exercises, use the precise definition of limit to prove the limit.<\/p>\n

      \n
      \n

      21.\u00a0<\/strong>[latex]\\underset{x\\to 1}{\\lim}(8x+16)=24[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      22.\u00a0<\/strong>[latex]\\underset{x\\to 0}{\\lim}x^3=0[\/latex]<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex]\\delta =\\sqrt[3]{\\epsilon}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      23.\u00a0<\/strong>A ball is thrown into the air and the vertical position is given by [latex]x(t)=-4.9t^2+25t+5[\/latex]. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      24.\u00a0<\/strong>A particle moving along a line has a displacement according to the function [latex]x(t)=t^2-2t+4[\/latex], where [latex]x[\/latex] is measured in meters and [latex]t[\/latex] is measured in seconds. Find the average velocity over the time period [latex]t=[0,2][\/latex].<\/p>\n<\/div>\n

      \n
      Show Solution<\/span><\/p>\n
      \n

      [latex]0[\/latex] m\/sec<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \n
      \n

      25.\u00a0<\/strong>From the previous exercises, estimate the instantaneous velocity at [latex]t=2[\/latex] by checking the average velocity within [latex]t=0.01[\/latex] sec.<\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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      Candela Citations<\/h3>\n\t\t\t\t\t
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      CC licensed content, Shared previously<\/div>