{"id":2849,"date":"2025-08-13T20:18:16","date_gmt":"2025-08-13T20:18:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2849"},"modified":"2025-12-02T22:49:24","modified_gmt":"2025-12-02T22:49:24","slug":"finding-limits-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/finding-limits-learn-it-3\/","title":{"raw":"Finding Limits: Learn It 3","rendered":"Finding Limits: Learn It 3"},"content":{"raw":"

Finding a Limit Using a Graph<\/h2>\r\nTo visually determine if a limit exists as [latex]x[\/latex] approaches [latex]a[\/latex], we observe the graph of the function when [latex]x[\/latex] is very near to [latex]x=a[\/latex]. We observe the behavior of the graph on both sides of [latex]a[\/latex].\r\n\r\n\"Graph\r\n\r\nTo determine if a left-hand limit exists, we observe the branch of the graph to the left of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x<a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex] so there is a left-hand limit.\r\n\r\nTo determine if a right-hand limit exists, observe the branch of the graph to the right of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]x>a[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex], so there is a right-hand limit.\r\n\r\nIf the left-hand limit and the right-hand limit are the same, then we know that the function has a two-sided limit. Normally, when we refer to a \"limit,\" we mean a two-sided limit, unless we call it a one-sided limit.\r\n\r\nFinally, we can look for an output value for the function [latex]f\\left(x\\right)[\/latex] when the input value [latex]x[\/latex] is equal to [latex]a[\/latex]. The coordinate pair of the point would be [latex]\\left(a,f\\left(a\\right)\\right)[\/latex]. If such a point exists, then [latex]f\\left(a\\right)[\/latex] has a value. If the point does not exist, then we say that [latex]f\\left(a\\right)[\/latex] does not exist.\r\n\r\n
How To: Given a function [latex]f\\left(x\\right)[\/latex], use a graph to find the limits and a function value as [latex]x[\/latex] approaches [latex]a[\/latex].<\/strong>\r\n
    \r\n \t
  1. Examine the graph to determine whether a left-hand limit exists.<\/li>\r\n \t
  2. Examine the graph to determine whether a right-hand limit exists.<\/li>\r\n \t
  3. If the two one-sided limits exist and are equal, then there is a two-sided limit\u2014what we normally call a \"limit.\"<\/li>\r\n \t
  4. If there is a point at [latex]x=a[\/latex], then [latex]f\\left(a\\right)[\/latex] is the corresponding function value.<\/li>\r\n<\/ol>\r\n<\/section>
    \r\n
      \r\n \t
    1. Use the graph to answer the following questions.\"Graph\r\n
        \r\n \t
      1. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
      2. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
      3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
      4. [latex]f\\left(2\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      5. Use the graph to answer the following questions.\"Graph\r\n
          \r\n \t
        1. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
        2. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
        3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\r\n \t
        4. [latex]f\\left(2\\right)[\/latex]<\/li>\r\n<\/ol>\r\n
          <\/div>\r\n[reveal-answer q=\"459090\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"459090\"]\r\n
            \r\n \t
          1. \r\n
              \r\n \t
            1. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x<2[\/latex], but infinitesimally close to 2, the output values get close to [latex]y=8[\/latex].<\/li>\r\n \t
            2. [latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=3[\/latex]; when [latex]x>2[\/latex], but infinitesimally close to 2, the output values approach [latex]y=3[\/latex].<\/li>\r\n \t
            3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]; the left and right-hand limits are not equal.<\/li>\r\n \t
            4. [latex]f\\left(2\\right)=3[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,3\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
            5. \r\n
                \r\n \t
              1. [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x<2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\r\n \t
              2. [latex]\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; when [latex]x>2[\/latex] but infinitesimally close to 2, the output values approach [latex]y=8[\/latex].<\/li>\r\n \t
              3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex] because [latex]\\underset{x\\to 2{}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to 2{}^{+}}{\\mathrm{lim}}f\\left(x\\right)=8[\/latex]; the left and right-hand limits are equal.<\/li>\r\n \t
              4. [latex]f\\left(2\\right)=4[\/latex] because the graph of the function [latex]f[\/latex] passes through the point [latex]\\left(2,f\\left(2\\right)\\right)[\/latex] or [latex]\\left(2,4\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/section>
                Using the graph of the function [latex]y=f\\left(x\\right)[\/latex], estimate the following limits.\"Graph[reveal-answer q=\"319433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"319433\"]a. 0; b. 2; c. does not exist; d. [latex]-2[\/latex]; e. 0; f. does not exist; g. 4; h. 4; i. 4\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
                [ohm_question hide_question_numbers=1]174077[\/ohm_question]<\/section>","rendered":"

                Finding a Limit Using a Graph<\/h2>\n

                To visually determine if a limit exists as [latex]x[\/latex] approaches [latex]a[\/latex], we observe the graph of the function when [latex]x[\/latex] is very near to [latex]x=a[\/latex]. We observe the behavior of the graph on both sides of [latex]a[\/latex].<\/p>\n

                \"Graph<\/p>\n

                To determine if a left-hand limit exists, we observe the branch of the graph to the left of [latex]x=a[\/latex], but near [latex]x=a[\/latex]. This is where [latex]xa[\/latex]. We see that the outputs are getting close to some real number [latex]L[\/latex], so there is a right-hand limit.<\/p>\n

                If the left-hand limit and the right-hand limit are the same, then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.<\/p>\n

                Finally, we can look for an output value for the function [latex]f\\left(x\\right)[\/latex] when the input value [latex]x[\/latex] is equal to [latex]a[\/latex]. The coordinate pair of the point would be [latex]\\left(a,f\\left(a\\right)\\right)[\/latex]. If such a point exists, then [latex]f\\left(a\\right)[\/latex] has a value. If the point does not exist, then we say that [latex]f\\left(a\\right)[\/latex] does not exist.<\/p>\n

                How To: Given a function [latex]f\\left(x\\right)[\/latex], use a graph to find the limits and a function value as [latex]x[\/latex] approaches [latex]a[\/latex].<\/strong><\/p>\n
                  \n
                1. Examine the graph to determine whether a left-hand limit exists.<\/li>\n
                2. Examine the graph to determine whether a right-hand limit exists.<\/li>\n
                3. If the two one-sided limits exist and are equal, then there is a two-sided limit\u2014what we normally call a “limit.”<\/li>\n
                4. If there is a point at [latex]x=a[\/latex], then [latex]f\\left(a\\right)[\/latex] is the corresponding function value.<\/li>\n<\/ol>\n<\/section>\n
                  \n
                    \n
                  1. Use the graph to answer the following questions.\"Graph\n
                      \n
                    1. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                    2. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                    3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                    4. [latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                    5. Use the graph to answer the following questions.\"Graph\n
                        \n
                      1. [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                      2. [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                      3. [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/li>\n
                      4. [latex]f\\left(2\\right)[\/latex]<\/li>\n<\/ol>\n
                        <\/div>\n