{"id":2002,"date":"2024-05-10T16:30:22","date_gmt":"2024-05-10T16:30:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=2002"},"modified":"2024-08-05T01:41:09","modified_gmt":"2024-08-05T01:41:09","slug":"limits-and-continuity-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-and-continuity-get-stronger\/","title":{"raw":"Limits and Continuity: Get Stronger","rendered":"Limits and Continuity: Get Stronger"},"content":{"raw":"

The Limit Laws<\/h2>\r\n

In the following exercises (1-2), use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).<\/strong><\/p>\r\n

    \r\n\t
  1. [latex]\\underset{x\\to 0}{\\lim}(4x^2-2x+3)[\/latex]<\/li>\r\n\t
  2. [latex]\\underset{x\\to -2}{\\lim}\\sqrt{x^2-6x+3}[\/latex]<\/li>\r\n<\/ol>\r\n

    In the following exercises (3-5), use direct substitution to evaluate each limit.<\/strong><\/p>\r\n

      \r\n\t
    1. [latex]\\underset{x\\to 7}{\\lim}x^2[\/latex]<\/li>\r\n\t
    2. [latex]\\underset{x\\to 0}{\\lim}\\dfrac{1}{1+ \\sin x}[\/latex]<\/li>\r\n\t
    3. [latex]\\underset{x\\to 1}{\\lim}\\dfrac{2-7x}{x+6}[\/latex]<\/li>\r\n<\/ol>\r\n

      In the following exercises (6-10), use direct substitution to show that each limit leads to the indeterminate form [latex]\\frac{0}{0}[\/latex]. Then, evaluate the limit.<\/strong><\/p>\r\n

        \r\n\t
      1. [latex]\\underset{x\\to 4}{\\lim}\\dfrac{x^2-16}{x-4}[\/latex]<\/li>\r\n\t
      2. [latex]\\underset{x\\to 6}{\\lim}\\dfrac{3x-18}{2x-12}[\/latex]<\/li>\r\n\t
      3. [latex]\\underset{t\\to 9}{\\lim}\\dfrac{t-9}{\\sqrt{t}-3}[\/latex]<\/li>\r\n\t
      4. [latex]\\underset{\\theta \\to \\pi}{\\lim}\\dfrac{\\sin \\theta}{\\tan \\theta}[\/latex]\r\n\r\n\r\n
        \u00a0<\/div>\r\n<\/li>\r\n\t
      5. [latex]\\underset{x\\to 1\/2}{\\lim}\\dfrac{2x^2+3x-2}{2x-1}[\/latex]<\/li>\r\n<\/ol>\r\n

        In the following exercises (11-14), assume that [latex]\\underset{x\\to 6}{\\lim}f(x)=4, \\, \\underset{x\\to 6}{\\lim}g(x)=9[\/latex], and [latex]\\underset{x\\to 6}{\\lim}h(x)=6[\/latex]. Use these three facts and the limit laws to evaluate each limit.<\/strong><\/p>\r\n

          \r\n\t
        1. [latex]\\underset{x\\to 6}{\\lim}2f(x)g(x)[\/latex]<\/li>\r\n\t
        2. [latex]\\underset{x\\to 6}{\\lim}(f(x)+\\frac{1}{3}g(x))[\/latex]<\/li>\r\n\t
        3. [latex]\\underset{x\\to 6}{\\lim}\\sqrt{g(x)-f(x)}[\/latex]<\/li>\r\n\t
        4. [latex]\\underset{x\\to 6}{\\lim}[(x+1)\\cdot f(x)][\/latex]<\/li>\r\n<\/ol>\r\n

          In the following exercises (15-16), use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.<\/strong><\/p>\r\n

            \r\n\t
          1. [latex]f(x)=\\begin{cases} x^2, & x \\le 3 \\\\ x+4, & x > 3 \\end{cases}[\/latex]
            \r\n
              \r\n\t
            1. [latex]\\underset{x\\to 3^-}{\\lim}f(x)[\/latex]<\/li>\r\n\t
            2. [latex]\\underset{x\\to 3^+}{\\lim}f(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t
            3. [latex]h(x)=\\begin{cases} x^2-2x+1, & x < 2 \\\\ 3 - x, & x \\ge 2 \\end{cases}[\/latex]\r\n\r\n\r\n
                \r\n\t
              1. [latex]\\underset{x\\to 2^-}{\\lim}h(x)[\/latex]<\/li>\r\n\t
              2. [latex]\\underset{x\\to 2^+}{\\lim}h(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

                In the following exercises (17-20), use the graphs below and the limit laws to evaluate each limit.<\/strong><\/p>\r\n

                \"Two<\/span><\/p>\r\n

                  \r\n\t
                1. [latex]\\underset{x\\to -3^-}{\\lim}(f(x)-3g(x))[\/latex]<\/li>\r\n\t
                2. [latex]\\underset{x\\to -5}{\\lim}\\dfrac{2+g(x)}{f(x)}[\/latex]<\/li>\r\n\t
                3. [latex]\\underset{x\\to 1}{\\lim}\\sqrt[3]{f(x)-g(x)}[\/latex]<\/li>\r\n\t
                4. [latex]\\underset{x\\to -9}{\\lim}[xf(x)+2g(x)][\/latex]<\/li>\r\n<\/ol>\r\n

                  Continuity<\/h2>\r\n

                  For the following exercises (1-4), determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.<\/strong><\/p>\r\n

                    \r\n\t
                  1. [latex]f(x)=\\dfrac{1}{\\sqrt{x}}[\/latex]<\/li>\r\n\t
                  2. [latex]f(x)=\\dfrac{x}{x^2-x}[\/latex]<\/li>\r\n\t
                  3. [latex]f(x)=\\dfrac{5}{e^x-2}[\/latex]<\/li>\r\n\t
                  4. [latex]H(x)= \\tan 2x[\/latex]<\/li>\r\n<\/ol>\r\n

                    For the following exercises (5-7), decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?<\/strong><\/p>\r\n

                      \r\n\t
                    1. [latex]f(x)=\\dfrac{2x^2-5x+3}{x-1}[\/latex] at [latex]x=1[\/latex]<\/li>\r\n\t
                    2. [latex]g(u)=\\begin{cases} \\dfrac{6u^2+u-2}{2u-1} & \\text{ if } \\, u \\ne \\frac{1}{2} \\\\ \\dfrac{7}{2} & \\text{ if } \\, u = \\frac{1}{2} \\end{cases}[\/latex] at [latex]u=\\frac{1}{2}[\/latex]<\/li>\r\n\t
                    3. [latex]f(x)=\\begin{cases} x^2-e^x & \\text{ if } \\, x < 0 \\\\ x-1 & \\text{ if } \\, x \\ge 0 \\end{cases}[\/latex] at [latex]x=0[\/latex]<\/li>\r\n<\/ol>\r\n

                      In the following exercises (8-10), find the value(s) of [latex]k[\/latex] that makes each function continuous over the given interval.<\/strong><\/p>\r\n

                        \r\n\t
                      1. [latex]f(x)=\\begin{cases} 3x+2 & \\text{ if } \\, x < k \\\\ 2x-3 & \\text{ if } \\, k \\le x \\le 8 \\end{cases}[\/latex]<\/li>\r\n\t
                      2. [latex]f(x)=\\begin{cases} \\dfrac{x^2+3x+2}{x+2} & \\text{ if } \\, x \\ne -2 \\\\ k & \\text{ if } \\, x = -2 \\end{cases}[\/latex]<\/li>\r\n\t
                      3. [latex]f(x)=\\begin{cases} \\sqrt{kx} & \\text{ if } \\, 0 \\le x \\le 3 \\\\ x+1 & \\text{ if } \\, 3 < x \\le 10 \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\n

                        In the following exercises (11-12), sketch the graphs.<\/strong><\/p>\r\n

                          \r\n\t
                        1. Let [latex]f(x)=\\begin{cases} 3x & \\text{ if } \\, x > 1 \\\\ x^3 & \\text{ if } \\, x < 1 \\end{cases}[\/latex]\r\n\r\n\r\n
                            \r\n\t
                          1. Sketch the graph of [latex]f[\/latex].<\/li>\r\n\t
                          2. Is it possible to find a value [latex]k[\/latex] such that [latex]f(1)=k[\/latex], which makes [latex]f(x)[\/latex] continuous for all real numbers? Briefly explain.<\/li>\r\n<\/ol>\r\n<\/li>\r\n\t
                          3. Sketch the graph of the function [latex]y=f(x)[\/latex] with properties 1 through 7.\r\n\r\n\r\n
                              \r\n\t
                            1. The domain of [latex]f[\/latex] is [latex](\u2212\\infty,+\\infty)[\/latex].<\/li>\r\n\t
                            2. [latex]f[\/latex] has an infinite discontinuity at [latex]x=-6[\/latex].<\/li>\r\n\t
                            3. [latex]f(-6)=3[\/latex]<\/li>\r\n\t
                            4. [latex]\\underset{x\\to -3^-}{\\lim}f(x)=\\underset{x\\to -3^+}{\\lim}f(x)=2[\/latex]<\/li>\r\n\t
                            5. [latex]f(-3)=3[\/latex]<\/li>\r\n\t
                            6. [latex]f[\/latex] is left continuous but not right continuous at [latex]x=3[\/latex].<\/li>\r\n\t
                            7. [latex]\\underset{x\\to -\\infty}{\\lim}f(x)=\u2212\\infty[\/latex] and [latex]\\underset{x\\to +\\infty}{\\lim}f(x)=+\\infty[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

                              In the following exercise, suppose [latex]y=f(x)[\/latex] is defined for all [latex]x[\/latex]. For each description, sketch a graph with the indicated property.<\/strong><\/p>\r\n

                                \r\n\t
                              1. Discontinuous at [latex]x=1[\/latex] with [latex]\\underset{x\\to -1}{\\lim}f(x)=-1[\/latex] and [latex]\\underset{x\\to 2}{\\lim}f(x)=4[\/latex]<\/li>\r\n<\/ol>\r\n

                                Determine whether each of the given statements is true (14-17). Justify your responses with an explanation or counterexample.<\/strong><\/p>\r\n

                                  \r\n\t
                                1. [latex]f(t)=\\dfrac{2}{e^t-e^{-t}}[\/latex] is continuous everywhere.<\/li>\r\n\t
                                2. If a function is not continuous at a point, then it is not defined at that point.<\/li>\r\n\t
                                3. If [latex]f(x)[\/latex] is continuous such that [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, then [latex]f(x)=0[\/latex] has exactly one solution in [latex][a,b][\/latex].<\/li>\r\n\t
                                4. If [latex]f(x)[\/latex] is continuous everywhere and [latex]f(a), f(b)>0[\/latex], then there is no root of [latex]f(x)[\/latex] in the interval [latex][a,b][\/latex].<\/li>\r\n<\/ol>\r\n

                                  Prove the following functions are continuous everywhere (18-19).<\/strong><\/p>\r\n

                                    \r\n\t
                                  1. [latex]f(\\theta) = \\sin \\theta[\/latex]<\/li>\r\n\t
                                  2. Where is [latex]f(x)=\\begin{cases} 0 & \\text{ if } \\, x \\, \\text{is irrational} \\\\ 1 & \\text{ if } \\, x \\, \\text{is rational} \\end{cases}[\/latex] continuous?<\/li>\r\n<\/ol>\r\n

                                    The Precise Definition of a Limit<\/h2>\r\n

                                    In the following exercises (1-2), write the appropriate [latex]\\varepsilon[\/latex]-[latex]\\delta[\/latex] definition for each of the given statements.<\/strong><\/p>\r\n

                                      \r\n\t
                                    1. [latex]\\underset{t\\to b}{\\lim}g(t)=M[\/latex] <\/li>\r\n\t
                                    2. [latex]\\underset{x\\to a}{\\lim}\\phi(x)=A[\/latex]\r\n\r\n\r\n
                                      \r\n<\/li>\r\n<\/ol>\r\n

                                      The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 2}{\\lim}f(x)=2[\/latex]. In the following exercise, determine a value of [latex]\\delta >0[\/latex] that satisfies the statement.<\/strong><\/p>\r\n

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

                                        \r\n\t
                                      1. If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<0.5[\/latex].<\/li>\r\n<\/ol>\r\n

                                        The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=-1[\/latex]. In the following exercise, determine a value of [latex]\\delta >0[\/latex] that satisfies the statement.<\/strong><\/p>\r\n

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

                                          \r\n\t
                                        1. If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<2[\/latex]. <\/li>\r\n<\/ol>\r\n

                                          The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=2[\/latex]. In the following exercise, for the given value of [latex]\\varepsilon[\/latex], find a value of [latex]\\delta >0[\/latex] such that the precise definition of limit holds true.<\/strong><\/p>\r\n

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

                                            \r\n\t
                                          1. [latex]\\varepsilon =3[\/latex] <\/li>\r\n<\/ol>\r\n

                                            In the following exercise, use a graphing calculator to find a number [latex]\\delta[\/latex] such that the statement holds true.<\/strong><\/p>\r\n

                                              \r\n\t
                                            1. [latex]|\\sqrt{x-4}-2|<0.1[\/latex], whenever [latex]|x-8|<\\delta [\/latex]<\/li>\r\n<\/ol>\r\n

                                              In the following exercises (7-8), use the precise definition of limit to prove the given limits.<\/strong><\/p>\r\n

                                                \r\n\t
                                              1. [latex]\\underset{x\\to 3}{\\lim}\\dfrac{x^2-9}{x-3}=6[\/latex] <\/li>\r\n\t
                                              2. [latex]\\underset{x\\to 0}{\\lim}x^4=0[\/latex]<\/li>\r\n<\/ol>\r\n

                                                In the following exercises (9-10), use the precise definition of limit to prove the given one-sided limits.<\/strong><\/p>\r\n

                                                  \r\n\t
                                                1. [latex]\\underset{x\\to 5^-}{\\lim}\\sqrt{5-x}=0[\/latex] <\/li>\r\n\t
                                                2. [latex]\\underset{x\\to 1^-}{\\lim}f(x)=3[\/latex], where [latex]f(x)=\\begin{cases} 5x-2 & \\text{ if } \\, x < 1 \\\\ 7x-1 & \\text{ if } x \\ge 1 \\end{cases}[\/latex] <\/li>\r\n<\/ol>\r\n

                                                  In the following exercise, use the precise definition of limit to prove the given infinite limit.<\/strong><\/p>\r\n

                                                    \r\n\t
                                                  1. [latex]\\underset{x\\to -1}{\\lim}\\dfrac{3}{(x+1)^2}=\\infty [\/latex] <\/li>\r\n<\/ol>\r\n

                                                    For the following exercises (12-13), suppose that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex] both exist. Use the precise definition of limits to prove the following limit laws:<\/strong><\/p>\r\n

                                                      \r\n\t
                                                    1. [latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=L-M[\/latex] <\/li>\r\n\t
                                                    2. [latex]\\underset{x\\to a}{\\lim}[f(x)g(x)]=LM[\/latex]. <\/li>\r\n<\/ol>","rendered":"

                                                      The Limit Laws<\/h2>\n

                                                      In the following exercises (1-2), use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).<\/strong><\/p>\n

                                                        \n
                                                      1. [latex]\\underset{x\\to 0}{\\lim}(4x^2-2x+3)[\/latex]<\/li>\n
                                                      2. [latex]\\underset{x\\to -2}{\\lim}\\sqrt{x^2-6x+3}[\/latex]<\/li>\n<\/ol>\n

                                                        In the following exercises (3-5), use direct substitution to evaluate each limit.<\/strong><\/p>\n

                                                          \n
                                                        1. [latex]\\underset{x\\to 7}{\\lim}x^2[\/latex]<\/li>\n
                                                        2. [latex]\\underset{x\\to 0}{\\lim}\\dfrac{1}{1+ \\sin x}[\/latex]<\/li>\n
                                                        3. [latex]\\underset{x\\to 1}{\\lim}\\dfrac{2-7x}{x+6}[\/latex]<\/li>\n<\/ol>\n

                                                          In the following exercises (6-10), use direct substitution to show that each limit leads to the indeterminate form [latex]\\frac{0}{0}[\/latex]. Then, evaluate the limit.<\/strong><\/p>\n

                                                            \n
                                                          1. [latex]\\underset{x\\to 4}{\\lim}\\dfrac{x^2-16}{x-4}[\/latex]<\/li>\n
                                                          2. [latex]\\underset{x\\to 6}{\\lim}\\dfrac{3x-18}{2x-12}[\/latex]<\/li>\n
                                                          3. [latex]\\underset{t\\to 9}{\\lim}\\dfrac{t-9}{\\sqrt{t}-3}[\/latex]<\/li>\n
                                                          4. [latex]\\underset{\\theta \\to \\pi}{\\lim}\\dfrac{\\sin \\theta}{\\tan \\theta}[\/latex]\n
                                                            \u00a0<\/div>\n<\/li>\n
                                                          5. [latex]\\underset{x\\to 1\/2}{\\lim}\\dfrac{2x^2+3x-2}{2x-1}[\/latex]<\/li>\n<\/ol>\n

                                                            In the following exercises (11-14), assume that [latex]\\underset{x\\to 6}{\\lim}f(x)=4, \\, \\underset{x\\to 6}{\\lim}g(x)=9[\/latex], and [latex]\\underset{x\\to 6}{\\lim}h(x)=6[\/latex]. Use these three facts and the limit laws to evaluate each limit.<\/strong><\/p>\n

                                                              \n
                                                            1. [latex]\\underset{x\\to 6}{\\lim}2f(x)g(x)[\/latex]<\/li>\n
                                                            2. [latex]\\underset{x\\to 6}{\\lim}(f(x)+\\frac{1}{3}g(x))[\/latex]<\/li>\n
                                                            3. [latex]\\underset{x\\to 6}{\\lim}\\sqrt{g(x)-f(x)}[\/latex]<\/li>\n
                                                            4. [latex]\\underset{x\\to 6}{\\lim}[(x+1)\\cdot f(x)][\/latex]<\/li>\n<\/ol>\n

                                                              In the following exercises (15-16), use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.<\/strong><\/p>\n

                                                                \n
                                                              1. [latex]f(x)=\\begin{cases} x^2, & x \\le 3 \\\\ x+4, & x > 3 \\end{cases}[\/latex]\n
                                                                  \n
                                                                1. [latex]\\underset{x\\to 3^-}{\\lim}f(x)[\/latex]<\/li>\n
                                                                2. [latex]\\underset{x\\to 3^+}{\\lim}f(x)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                3. [latex]h(x)=\\begin{cases} x^2-2x+1, & x < 2 \\\\ 3 - x, & x \\ge 2 \\end{cases}[\/latex]\n\n\n\n\n
                                                                    \n
                                                                  1. [latex]\\underset{x\\to 2^-}{\\lim}h(x)[\/latex]<\/li>\n
                                                                  2. [latex]\\underset{x\\to 2^+}{\\lim}h(x)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                    In the following exercises (17-20), use the graphs below and the limit laws to evaluate each limit.<\/strong><\/p>\n

                                                                    \"Two<\/span><\/p>\n

                                                                      \n
                                                                    1. [latex]\\underset{x\\to -3^-}{\\lim}(f(x)-3g(x))[\/latex]<\/li>\n
                                                                    2. [latex]\\underset{x\\to -5}{\\lim}\\dfrac{2+g(x)}{f(x)}[\/latex]<\/li>\n
                                                                    3. [latex]\\underset{x\\to 1}{\\lim}\\sqrt[3]{f(x)-g(x)}[\/latex]<\/li>\n
                                                                    4. [latex]\\underset{x\\to -9}{\\lim}[xf(x)+2g(x)][\/latex]<\/li>\n<\/ol>\n

                                                                      Continuity<\/h2>\n

                                                                      For the following exercises (1-4), determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.<\/strong><\/p>\n

                                                                        \n
                                                                      1. [latex]f(x)=\\dfrac{1}{\\sqrt{x}}[\/latex]<\/li>\n
                                                                      2. [latex]f(x)=\\dfrac{x}{x^2-x}[\/latex]<\/li>\n
                                                                      3. [latex]f(x)=\\dfrac{5}{e^x-2}[\/latex]<\/li>\n
                                                                      4. [latex]H(x)= \\tan 2x[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercises (5-7), decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?<\/strong><\/p>\n

                                                                          \n
                                                                        1. [latex]f(x)=\\dfrac{2x^2-5x+3}{x-1}[\/latex] at [latex]x=1[\/latex]<\/li>\n
                                                                        2. [latex]g(u)=\\begin{cases} \\dfrac{6u^2+u-2}{2u-1} & \\text{ if } \\, u \\ne \\frac{1}{2} \\\\ \\dfrac{7}{2} & \\text{ if } \\, u = \\frac{1}{2} \\end{cases}[\/latex] at [latex]u=\\frac{1}{2}[\/latex]<\/li>\n
                                                                        3. [latex]f(x)=\\begin{cases} x^2-e^x & \\text{ if } \\, x < 0 \\\\ x-1 & \\text{ if } \\, x \\ge 0 \\end{cases}[\/latex] at [latex]x=0[\/latex]<\/li>\n<\/ol>\n

                                                                          In the following exercises (8-10), find the value(s) of [latex]k[\/latex] that makes each function continuous over the given interval.<\/strong><\/p>\n

                                                                            \n
                                                                          1. [latex]f(x)=\\begin{cases} 3x+2 & \\text{ if } \\, x < k \\\\ 2x-3 & \\text{ if } \\, k \\le x \\le 8 \\end{cases}[\/latex]<\/li>\n
                                                                          2. [latex]f(x)=\\begin{cases} \\dfrac{x^2+3x+2}{x+2} & \\text{ if } \\, x \\ne -2 \\\\ k & \\text{ if } \\, x = -2 \\end{cases}[\/latex]<\/li>\n
                                                                          3. [latex]f(x)=\\begin{cases} \\sqrt{kx} & \\text{ if } \\, 0 \\le x \\le 3 \\\\ x+1 & \\text{ if } \\, 3 < x \\le 10 \\end{cases}[\/latex]<\/li>\n<\/ol>\n

                                                                            In the following exercises (11-12), sketch the graphs.<\/strong><\/p>\n

                                                                              \n
                                                                            1. Let [latex]f(x)=\\begin{cases} 3x & \\text{ if } \\, x > 1 \\\\ x^3 & \\text{ if } \\, x < 1 \\end{cases}[\/latex]\n\n\n\n\n
                                                                                \n
                                                                              1. Sketch the graph of [latex]f[\/latex].<\/li>\n
                                                                              2. Is it possible to find a value [latex]k[\/latex] such that [latex]f(1)=k[\/latex], which makes [latex]f(x)[\/latex] continuous for all real numbers? Briefly explain.<\/li>\n<\/ol>\n<\/li>\n
                                                                              3. Sketch the graph of the function [latex]y=f(x)[\/latex] with properties 1 through 7.\n
                                                                                  \n
                                                                                1. The domain of [latex]f[\/latex] is [latex](\u2212\\infty,+\\infty)[\/latex].<\/li>\n
                                                                                2. [latex]f[\/latex] has an infinite discontinuity at [latex]x=-6[\/latex].<\/li>\n
                                                                                3. [latex]f(-6)=3[\/latex]<\/li>\n
                                                                                4. [latex]\\underset{x\\to -3^-}{\\lim}f(x)=\\underset{x\\to -3^+}{\\lim}f(x)=2[\/latex]<\/li>\n
                                                                                5. [latex]f(-3)=3[\/latex]<\/li>\n
                                                                                6. [latex]f[\/latex] is left continuous but not right continuous at [latex]x=3[\/latex].<\/li>\n
                                                                                7. [latex]\\underset{x\\to -\\infty}{\\lim}f(x)=\u2212\\infty[\/latex] and [latex]\\underset{x\\to +\\infty}{\\lim}f(x)=+\\infty[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                  In the following exercise, suppose [latex]y=f(x)[\/latex] is defined for all [latex]x[\/latex]. For each description, sketch a graph with the indicated property.<\/strong><\/p>\n

                                                                                    \n
                                                                                  1. Discontinuous at [latex]x=1[\/latex] with [latex]\\underset{x\\to -1}{\\lim}f(x)=-1[\/latex] and [latex]\\underset{x\\to 2}{\\lim}f(x)=4[\/latex]<\/li>\n<\/ol>\n

                                                                                    Determine whether each of the given statements is true (14-17). Justify your responses with an explanation or counterexample.<\/strong><\/p>\n

                                                                                      \n
                                                                                    1. [latex]f(t)=\\dfrac{2}{e^t-e^{-t}}[\/latex] is continuous everywhere.<\/li>\n
                                                                                    2. If a function is not continuous at a point, then it is not defined at that point.<\/li>\n
                                                                                    3. If [latex]f(x)[\/latex] is continuous such that [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, then [latex]f(x)=0[\/latex] has exactly one solution in [latex][a,b][\/latex].<\/li>\n
                                                                                    4. If [latex]f(x)[\/latex] is continuous everywhere and [latex]f(a), f(b)>0[\/latex], then there is no root of [latex]f(x)[\/latex] in the interval [latex][a,b][\/latex].<\/li>\n<\/ol>\n

                                                                                      Prove the following functions are continuous everywhere (18-19).<\/strong><\/p>\n

                                                                                        \n
                                                                                      1. [latex]f(\\theta) = \\sin \\theta[\/latex]<\/li>\n
                                                                                      2. Where is [latex]f(x)=\\begin{cases} 0 & \\text{ if } \\, x \\, \\text{is irrational} \\\\ 1 & \\text{ if } \\, x \\, \\text{is rational} \\end{cases}[\/latex] continuous?<\/li>\n<\/ol>\n

                                                                                        The Precise Definition of a Limit<\/h2>\n

                                                                                        In the following exercises (1-2), write the appropriate [latex]\\varepsilon[\/latex]–[latex]\\delta[\/latex] definition for each of the given statements.<\/strong><\/p>\n

                                                                                          \n
                                                                                        1. [latex]\\underset{t\\to b}{\\lim}g(t)=M[\/latex] <\/li>\n
                                                                                        2. [latex]\\underset{x\\to a}{\\lim}\\phi(x)=A[\/latex]\n
                                                                                          \n<\/div>\n<\/li>\n<\/ol>\n

                                                                                          The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 2}{\\lim}f(x)=2[\/latex]. In the following exercise, determine a value of [latex]\\delta >0[\/latex] that satisfies the statement.<\/strong><\/p>\n

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

                                                                                            \n
                                                                                          1. If [latex]0<|x-2|<\\delta[\/latex], then [latex]|f(x)-2|<0.5[\/latex].<\/li>\n<\/ol>\n

                                                                                            The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=-1[\/latex]. In the following exercise, determine a value of [latex]\\delta >0[\/latex] that satisfies the statement.<\/strong><\/p>\n

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

                                                                                              \n
                                                                                            1. If [latex]0<|x-3|<\\delta[\/latex], then [latex]|f(x)+1|<2[\/latex]. <\/li>\n<\/ol>\n

                                                                                              The following graph of the function [latex]f[\/latex] satisfies [latex]\\underset{x\\to 3}{\\lim}f(x)=2[\/latex]. In the following exercise, for the given value of [latex]\\varepsilon[\/latex], find a value of [latex]\\delta >0[\/latex] such that the precise definition of limit holds true.<\/strong><\/p>\n

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

                                                                                                \n
                                                                                              1. [latex]\\varepsilon =3[\/latex] <\/li>\n<\/ol>\n

                                                                                                In the following exercise, use a graphing calculator to find a number [latex]\\delta[\/latex] such that the statement holds true.<\/strong><\/p>\n

                                                                                                  \n
                                                                                                1. [latex]|\\sqrt{x-4}-2|<0.1[\/latex], whenever [latex]|x-8|<\\delta[\/latex]<\/li>\n<\/ol>\n

                                                                                                  In the following exercises (7-8), use the precise definition of limit to prove the given limits.<\/strong><\/p>\n

                                                                                                    \n
                                                                                                  1. [latex]\\underset{x\\to 3}{\\lim}\\dfrac{x^2-9}{x-3}=6[\/latex] <\/li>\n
                                                                                                  2. [latex]\\underset{x\\to 0}{\\lim}x^4=0[\/latex]<\/li>\n<\/ol>\n

                                                                                                    In the following exercises (9-10), use the precise definition of limit to prove the given one-sided limits.<\/strong><\/p>\n

                                                                                                      \n
                                                                                                    1. [latex]\\underset{x\\to 5^-}{\\lim}\\sqrt{5-x}=0[\/latex] <\/li>\n
                                                                                                    2. [latex]\\underset{x\\to 1^-}{\\lim}f(x)=3[\/latex], where [latex]f(x)=\\begin{cases} 5x-2 & \\text{ if } \\, x < 1 \\\\ 7x-1 & \\text{ if } x \\ge 1 \\end{cases}[\/latex] <\/li>\n<\/ol>\n

                                                                                                      In the following exercise, use the precise definition of limit to prove the given infinite limit.<\/strong><\/p>\n

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

                                                                                                        For the following exercises (12-13), suppose that [latex]\\underset{x\\to a}{\\lim}f(x)=L[\/latex] and [latex]\\underset{x\\to a}{\\lim}g(x)=M[\/latex] both exist. Use the precise definition of limits to prove the following limit laws:<\/strong><\/p>\n

                                                                                                          \n
                                                                                                        1. [latex]\\underset{x\\to a}{\\lim}(f(x)-g(x))=L-M[\/latex] <\/li>\n
                                                                                                        2. [latex]\\underset{x\\to a}{\\lim}[f(x)g(x)]=LM[\/latex]. <\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":21,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":185,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2002"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":27,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2002\/revisions"}],"predecessor-version":[{"id":2118,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2002\/revisions\/2118"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/2002\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=2002"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=2002"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=2002"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=2002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}