{"id":1064,"date":"2025-10-15T13:55:23","date_gmt":"2025-10-15T13:55:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1064"},"modified":"2025-10-23T13:49:21","modified_gmt":"2025-10-23T13:49:21","slug":"limits-and-continuity-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/limits-and-continuity-learn-it-2\/","title":{"raw":"Limits and Continuity: Learn It 2","rendered":"Limits and Continuity: Learn It 2"},"content":{"raw":"

Interior Points and Boundary Points<\/h2>\r\nTo study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.\r\n\r\n
\r\n

interior and boundary points<\/h3>\r\n

Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\r\n\r\n

    \r\n \t
  • A point [latex]P_0[\/latex] is an interior point<\/strong> of [latex]S[\/latex] if there is a [latex]\\delta[\/latex] disk centered around [latex]P_0[\/latex] contained completely in [latex]S[\/latex].<\/li>\r\n \t
  • A point [latex]P_0[\/latex] is a boundary point<\/strong> of [latex]S[\/latex] if every [latex]\\delta[\/latex] disk centered around [latex]P_0[\/latex] contains points both inside and outside [latex]S[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>[caption id=\"attachment_974\" align=\"aligncenter\" width=\"417\"]\"On Figure 4.\u00a0In the set [latex]S[\/latex] shown,\u00a0[latex]\\small{(-1,1)}[\/latex] is an interior point and\u00a0[latex]\\small{(2,3)}[\/latex]\u00a0is a boundary point.[\/caption]
    \r\n

    open and closed sets<\/h3>\r\n

    Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\r\n\r\n

      \r\n \t
    • [latex]S[\/latex] is an open set<\/strong> if every point of [latex]S[\/latex] is an interior point.<\/li>\r\n \t
    • [latex]S[\/latex] is a closed set<\/strong> if it contains all its boundary points.<\/li>\r\n<\/ul>\r\n<\/section>An example of an open set is a [latex]\\delta[\/latex] disk. If we include the boundary of the disk, then it becomes a closed set. A set that contains some, but not all, of its boundary points is neither open nor closed. For example, if we include half the boundary of a [latex]\\delta[\/latex] disk but not the other half, then the set is neither open nor closed.\r\n\r\n
      \r\n

      connected sets and regions<\/h3>\r\n

      Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\r\n\r\n

        \r\n \t
      • An open set [latex]S[\/latex] is connected<\/strong> if it cannot be represented as the union of two or more disjoint, nonempty open subsets.<\/li>\r\n \t
      • A set [latex]S[\/latex] is a region<\/strong> if it is open, connected, and nonempty.<\/li>\r\n<\/ul>\r\n<\/section>\r\n

        Limits at Boundary Points<\/h3>\r\n

        The definition of a limit of a function of two variables requires the [latex]\\delta[\/latex] disk to be contained inside the domain of the function. However, if we wish to find the limit of a function at a boundary point of the domain, the [latex]\\delta[\/latex] disk is not contained inside the domain. By definition, some points of the [latex]\\delta[\/latex] disk are inside the domain and some are outside.<\/p>\r\n

        Therefore, we need only consider points that are inside both the [latex]\\delta[\/latex] disk and the domain of the function. This leads to the definition of the limit of a function at a boundary point.<\/p>\r\n\r\n

        \r\n

        limit at a boundary point<\/h3>\r\n

        Let [latex]f[\/latex] be a function of two variables, and suppose [latex](a,\\ b)[\/latex] is on the boundary of the domain of [latex]f[\/latex]. Then<\/p>\r\n

        [latex]\\underset{(x,y)\\to(a,b)}{\\text{lim}}f(x,y)=L[\/latex]<\/p>\r\n

        if for any [latex]\\epsilon>0[\/latex], there exists [latex]\\delta>0[\/latex] such that<\/p>\r\n

        [latex]|f(x,y)-L|<\\epsilon[\/latex] whenever [latex]0<\\sqrt{(x-a)^{2}+(y-b)^{2}}<\\delta[\/latex]<\/p>\r\n

        for all points [latex](x,\\ y)[\/latex] inside the domain of [latex]f[\/latex].<\/p>\r\n\r\n<\/section>

        Prove [latex]\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}\\sqrt{25-x^{2}-y^{2}}=0[\/latex].[reveal-answer q=\"fs-id1667743933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667743933114\"]\r\n
        The domain of the function [latex]f\\,(x,\\ y)=\\sqrt{25-x^{2}-y^{2}}[\/latex] is [latex]\\{(x,\\ y)\\in\\mathbb{R}^{2}\\,|\\,x^{2}+y^{2}\\,\\leq\\,25\\}[\/latex], which\u00a0is a circle of radius [latex]5[\/latex]\u00a0centered at the origin, along with its interior as shown in the following graph.<\/div>\r\n
        [caption id=\"attachment_976\" align=\"aligncenter\" width=\"417\"]\"A Figure 5. Domain of the function [latex]\\small{f(x,y)=\\sqrt{25-x^{2}-y^{2}}}[\/latex].[\/caption]<\/div>\r\n
        We can\u00a0use the limit laws, which apply to limits at the boundary of domains as well as interior points:<\/div>\r\n
        [latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}\\sqrt{25-x^{2}-y^{2}}} & =\\hfill & {\\sqrt{\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}(25-x^{2}-y^{2})}} \\hfill \\\\ \\hfill & =\\hfill & {\\sqrt{\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}25-\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}x^{2}-\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}y^{2}}} \\hfill \\\\ \\hfill & =\\hfill & {\\sqrt{25-4^{2}-3^{2}}}\\hfill \\\\ \\hfill & =\\hfill & {0.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n
        See the following graph.<\/div>\r\n
        [caption id=\"attachment_977\" align=\"aligncenter\" width=\"353\"]\"The Figure 6.\u00a0Graph of the function [latex]\\small{f(x,y)=\\sqrt{25-x^{2}-y^{2}}}[\/latex].[\/caption]<\/div>\r\n
        <\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

        Interior Points and Boundary Points<\/h2>\n

        To study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.<\/p>\n

        \n

        interior and boundary points<\/h3>\n

        Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\n

          \n
        • A point [latex]P_0[\/latex] is an interior point<\/strong> of [latex]S[\/latex] if there is a [latex]\\delta[\/latex] disk centered around [latex]P_0[\/latex] contained completely in [latex]S[\/latex].<\/li>\n
        • A point [latex]P_0[\/latex] is a boundary point<\/strong> of [latex]S[\/latex] if every [latex]\\delta[\/latex] disk centered around [latex]P_0[\/latex] contains points both inside and outside [latex]S[\/latex]<\/li>\n<\/ul>\n<\/section>\n
          \"On
          Figure 4.\u00a0In the set [latex]S[\/latex] shown,\u00a0[latex]\\small{(-1,1)}[\/latex] is an interior point and\u00a0[latex]\\small{(2,3)}[\/latex]\u00a0is a boundary point.<\/figcaption><\/figure>\n
          \n

          open and closed sets<\/h3>\n

          Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\n

            \n
          • [latex]S[\/latex] is an open set<\/strong> if every point of [latex]S[\/latex] is an interior point.<\/li>\n
          • [latex]S[\/latex] is a closed set<\/strong> if it contains all its boundary points.<\/li>\n<\/ul>\n<\/section>\n

            An example of an open set is a [latex]\\delta[\/latex] disk. If we include the boundary of the disk, then it becomes a closed set. A set that contains some, but not all, of its boundary points is neither open nor closed. For example, if we include half the boundary of a [latex]\\delta[\/latex] disk but not the other half, then the set is neither open nor closed.<\/p>\n

            \n

            connected sets and regions<\/h3>\n

            Let [latex]S[\/latex] be a subset of [latex]\\mathbb{R}^{2}[\/latex].<\/p>\n

              \n
            • An open set [latex]S[\/latex] is connected<\/strong> if it cannot be represented as the union of two or more disjoint, nonempty open subsets.<\/li>\n
            • A set [latex]S[\/latex] is a region<\/strong> if it is open, connected, and nonempty.<\/li>\n<\/ul>\n<\/section>\n

              Limits at Boundary Points<\/h3>\n

              The definition of a limit of a function of two variables requires the [latex]\\delta[\/latex] disk to be contained inside the domain of the function. However, if we wish to find the limit of a function at a boundary point of the domain, the [latex]\\delta[\/latex] disk is not contained inside the domain. By definition, some points of the [latex]\\delta[\/latex] disk are inside the domain and some are outside.<\/p>\n

              Therefore, we need only consider points that are inside both the [latex]\\delta[\/latex] disk and the domain of the function. This leads to the definition of the limit of a function at a boundary point.<\/p>\n

              \n

              limit at a boundary point<\/h3>\n

              Let [latex]f[\/latex] be a function of two variables, and suppose [latex](a,\\ b)[\/latex] is on the boundary of the domain of [latex]f[\/latex]. Then<\/p>\n

              [latex]\\underset{(x,y)\\to(a,b)}{\\text{lim}}f(x,y)=L[\/latex]<\/p>\n

              if for any [latex]\\epsilon>0[\/latex], there exists [latex]\\delta>0[\/latex] such that<\/p>\n

              [latex]|f(x,y)-L|<\\epsilon[\/latex] whenever [latex]0<\\sqrt{(x-a)^{2}+(y-b)^{2}}<\\delta[\/latex]<\/p>\n

              for all points [latex](x,\\ y)[\/latex] inside the domain of [latex]f[\/latex].<\/p>\n<\/section>\n

              Prove [latex]\\displaystyle\\lim_{(x,\\ y)\\to(4,\\ 3)}\\sqrt{25-x^{2}-y^{2}}=0[\/latex].<\/p>\n