{"id":598,"date":"2025-10-09T15:24:38","date_gmt":"2025-10-09T15:24:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=598"},"modified":"2025-10-24T14:01:41","modified_gmt":"2025-10-24T14:01:41","slug":"limits-and-continuity-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/limits-and-continuity-fresh-take\/","title":{"raw":"Limits and Continuity: Fresh Take","rendered":"Limits and Continuity: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate the limit of a function of two variables<\/li>\r\n \t<li>Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach<\/li>\r\n \t<li>Determine when a function is continuous and check the conditions for continuity<\/li>\r\n \t<li>Find limits and check continuity for functions of three or more variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Limits<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Taking limits of functions with two variables follows similar rules to single-variable limits, but there's a crucial difference: you can approach a point from infinitely many directions.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Definition<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For [latex]\\underset{(x,y)\\to(a,b)}{\\text{lim}}f(x,y)=L[\/latex] to exist, [latex]f(x,y)[\/latex] must get arbitrarily close to [latex]L[\/latex] as [latex](x,y)[\/latex] approaches [latex](a,b)[\/latex]\u00a0from any direction.<\/p>\r\n<p class=\"whitespace-normal break-words\">We use a <strong>[latex]\\delta[\/latex] disk<\/strong>: a circle of radius [latex]\\delta[\/latex] around [latex](a,b)[\/latex]. All points in this disk (except possibly [latex](a,b)[\/latex]\u00a0itself) must give [latex]f[\/latex] values close to [latex]L[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Limit Laws<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The familiar limit laws extend to two variables:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Sum, difference, product, quotient laws<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Constant multiple, power, root laws<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity laws: [latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}x=a[\/latex] and [latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}y=b[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Use these laws to break down complex limits into simpler pieces.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>When Limits Don't Exist<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The limit must be the same along every possible path. If two different paths give different values, the limit doesn't exist.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Try substitution first using limit laws. If that fails, test different paths: try [latex]y=0[\/latex], [latex]x=0[\/latex], [latex]y=x[\/latex], or curves like [latex]y=x^2[\/latex]. If any two paths give different values, the limit doesn't exist.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following limit:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -2)}\\sqrt[3]{\\frac{x^{2}-y}{y^{2}+x-1}.}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1367793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367793933114\"]\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -2)}\\sqrt[3]{\\frac{x^{2}-y}{y^{2}+x-1}}=\\frac{3}{2}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Show that\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(2,\\ 1)}\\frac{(x-2)(y-1)}{(x-2)^{2}+(y-1)^{2}}[\/latex]<\/p>\r\ndoes not exist.\r\n\r\n[reveal-answer q=\"fs-id1667793933114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667793933114\"]\r\n<div style=\"text-align: center;\">If [latex]y=k\\,(x-2)+1[\/latex], then [latex]\\displaystyle\\lim_{(x,\\ y)\\to(2,\\ 1)}\\frac{(x-2)(y-1)}{(x-2)^{2}+(y-1)^{2}}=\\frac{k}{1+k^{2}}[\/latex]. Since the answer depends on [latex]k[\/latex], the limit fails to exist.<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186150&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pWvDsHUhTfY&amp;video_target=tpm-plugin-dfjtf1gg-pWvDsHUhTfY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.7_transcript.html\">transcript for \u201cCP 4.7\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Interior Points and Boundary Points<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">To understand continuity and limits in multivariable calculus, you need to distinguish between points inside a region and points on its edge.<\/p>\r\n<p class=\"whitespace-normal break-words\">A point [latex]P_0[\/latex] is an <strong>interior point<\/strong> of set [latex]S[\/latex] if you can draw a small disk around it that stays completely inside [latex]S[\/latex]. Think of it as having breathing room\u2014every nearby point is also in [latex]S[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">A point [latex]P_0[\/latex] is a <strong>boundary point<\/strong> of [latex]S[\/latex] if every disk around it contains both points inside [latex]S[\/latex] and points outside [latex]S[\/latex]. It sits on the edge.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Open and Closed Sets<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Open set<\/strong>: Every point is an interior point (no boundary included)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Closed set<\/strong>: Contains all its boundary points<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">A disk without its edge is open. A disk with its edge is closed. Include only part of the edge? Neither open nor closed.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Connected Sets and Regions<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Connected<\/strong>: Can't be split into separate pieces<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Region<\/strong>: Open, connected, and nonempty<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Limits at Boundary Points<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">When taking limits at boundary points, you only consider points inside both the [latex]\\delta[\/latex] disk and the domain. Some of the disk lies outside the domain\u2014that's fine. Just ignore those points.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following limit:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,-2)}\\sqrt{29-x^{2}-y^{2}}.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1667793733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667793733114\"]\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,-2)}\\sqrt{29-x^{2}-y^{2}}=0[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186151&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=zYJIT-NNVKM&amp;video_target=tpm-plugin-0y6dbuw0-zYJIT-NNVKM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.8_transcript.html\">transcript for \u201cCP 4.8\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Continuity<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">A function of two variables is continuous at a point when there are no jumps, breaks, or holes at that point. The definition mirrors single-variable continuity.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Three Conditions<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A function [latex]f(x, y)[\/latex] is continuous at [latex](a, b)[\/latex] if:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(a, b)[\/latex] exists<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f(x,y)[\/latex] exists<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f(x,y)=f(a,b)[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">All three must hold. The function must be defined, the limit must exist, and they must equal each other.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Checking Continuity<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Verify each condition:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Check the point is in the domain<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the limit using limit laws<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Confirm the limit equals the function value<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Properties of Continuous Functions<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Sum<\/strong>: If [latex]f[\/latex] and [latex]g[\/latex] are continuous, so is [latex]f + g[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Product<\/strong>: If [latex]g(x)[\/latex] and [latex]h(y)[\/latex] are continuous, so is [latex]g(x)h(y)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Composition<\/strong>: If [latex]g(x, y)[\/latex] is continuous and [latex]f(z)[\/latex] is continuous, so is [latex]f(g(x, y))[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Common Continuous Functions<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">Polynomials, rational functions (where denominators aren't zero), trig functions, exponentials, and logarithms are continuous on their domains. Compositions and combinations of these remain continuous.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Show that the function [latex]f\\,(x,y)=\\sqrt{26-2x^{2}-y^{2}}[\/latex]\u00a0is continuous at point [latex](2,-3)[\/latex].[reveal-answer q=\"fs-id1667993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667993733114\"]\r\n<ol>\r\n \t<li style=\"text-align: left;\">The domain of [latex]f[\/latex] contains the ordered pair [latex](2,-3)[\/latex] because [latex]f\\,(a,b)=f\\,(2,-3)=\\sqrt{16-2(2)^{2}-(-3)^{2}}=3[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=3[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=f\\,(a,b)=3[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Show that the functions [latex]f\\,(x,\\ y)=2x^{2}y^{3}+3[\/latex]\u00a0and [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex] are continuous everywhere.[reveal-answer q=\"fs-id1687993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1687993733114\"]\r\n\r\nThe polynomials [latex]g\\,(x)=2x^{2}[\/latex] and [latex]h\\,(y)=y^{3}[\/latex]\u00a0are continuous at every real number; therefore, by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\u00a0Furthermore, any constant function is continuous everywhere, so [latex]g\\,(x,\\ y)=3[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Therefore, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Last, [latex]h\\,(x)=x^{4}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0so by the continuity of composite functions theorem [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186152&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k4SalFB3XEw&amp;video_target=tpm-plugin-310u6ibm-k4SalFB3XEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.10_transcript.html\">transcript for \u201cCP 4.10\u201d here (opens in new window).<\/a><\/center><\/section>\r\n<h2>Functions of Three or More Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Limits and continuity extend naturally beyond two variables. The key is replacing [latex]\\delta[\/latex] disks with [latex]\\delta[\/latex] balls in higher dimensions.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The [latex]\\delta[\/latex] Ball<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">In three dimensions, a [latex]\\delta[\/latex] ball around [latex](x_0, y_0, z_0)[\/latex] includes all points within distance [latex]\\delta[\/latex]:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]{(x, y, z) \\mid \\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2} &lt; \\delta}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">For four dimensions, add another squared term under the radical. The pattern continues for any number of dimensions.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Limits in Three Variables<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">The definition mirrors two variables. For [latex]\\displaystyle\\lim_{(x,y,z)\\to(a,b,c)}f(x,y,z)=L[\/latex], points in a [latex]\\delta[\/latex] ball around [latex](a,b,c)[\/latex] must give function values arbitrarily close to [latex]L[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Using Limit Laws<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">All limit laws from two variables work for three or more variables:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Sum, difference, product, quotient<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Constant multiple, power, root<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identity laws<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Continuity<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">A function is continuous at a point if the same three conditions hold: the function exists, the limit exists, and they're equal. The concept is identical\u2014only the dimension changes.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}[\/latex].[reveal-answer q=\"fs-id1687993733119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1687993733119\"]\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}=2[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate the limit of a function of two variables<\/li>\n<li>Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach<\/li>\n<li>Determine when a function is continuous and check the conditions for continuity<\/li>\n<li>Find limits and check continuity for functions of three or more variables<\/li>\n<\/ul>\n<\/section>\n<h2>Limits<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Taking limits of functions with two variables follows similar rules to single-variable limits, but there&#8217;s a crucial difference: you can approach a point from infinitely many directions.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Definition<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For [latex]\\underset{(x,y)\\to(a,b)}{\\text{lim}}f(x,y)=L[\/latex] to exist, [latex]f(x,y)[\/latex] must get arbitrarily close to [latex]L[\/latex] as [latex](x,y)[\/latex] approaches [latex](a,b)[\/latex]\u00a0from any direction.<\/p>\n<p class=\"whitespace-normal break-words\">We use a <strong>[latex]\\delta[\/latex] disk<\/strong>: a circle of radius [latex]\\delta[\/latex] around [latex](a,b)[\/latex]. All points in this disk (except possibly [latex](a,b)[\/latex]\u00a0itself) must give [latex]f[\/latex] values close to [latex]L[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Limit Laws<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The familiar limit laws extend to two variables:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Sum, difference, product, quotient laws<\/li>\n<li class=\"whitespace-normal break-words\">Constant multiple, power, root laws<\/li>\n<li class=\"whitespace-normal break-words\">Identity laws: [latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}x=a[\/latex] and [latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}y=b[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Use these laws to break down complex limits into simpler pieces.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>When Limits Don&#8217;t Exist<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The limit must be the same along every possible path. If two different paths give different values, the limit doesn&#8217;t exist.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Try substitution first using limit laws. If that fails, test different paths: try [latex]y=0[\/latex], [latex]x=0[\/latex], [latex]y=x[\/latex], or curves like [latex]y=x^2[\/latex]. If any two paths give different values, the limit doesn&#8217;t exist.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following limit:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -2)}\\sqrt[3]{\\frac{x^{2}-y}{y^{2}+x-1}.}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1367793933114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1367793933114\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -2)}\\sqrt[3]{\\frac{x^{2}-y}{y^{2}+x-1}}=\\frac{3}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Show that<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(2,\\ 1)}\\frac{(x-2)(y-1)}{(x-2)^{2}+(y-1)^{2}}[\/latex]<\/p>\n<p>does not exist.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1667793933114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1667793933114\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">If [latex]y=k\\,(x-2)+1[\/latex], then [latex]\\displaystyle\\lim_{(x,\\ y)\\to(2,\\ 1)}\\frac{(x-2)(y-1)}{(x-2)^{2}+(y-1)^{2}}=\\frac{k}{1+k^{2}}[\/latex]. Since the answer depends on [latex]k[\/latex], the limit fails to exist.<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186150&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pWvDsHUhTfY&amp;video_target=tpm-plugin-dfjtf1gg-pWvDsHUhTfY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.7_transcript.html\">transcript for \u201cCP 4.7\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Interior Points and Boundary Points<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">To understand continuity and limits in multivariable calculus, you need to distinguish between points inside a region and points on its edge.<\/p>\n<p class=\"whitespace-normal break-words\">A point [latex]P_0[\/latex] is an <strong>interior point<\/strong> of set [latex]S[\/latex] if you can draw a small disk around it that stays completely inside [latex]S[\/latex]. Think of it as having breathing room\u2014every nearby point is also in [latex]S[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">A point [latex]P_0[\/latex] is a <strong>boundary point<\/strong> of [latex]S[\/latex] if every disk around it contains both points inside [latex]S[\/latex] and points outside [latex]S[\/latex]. It sits on the edge.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Open and Closed Sets<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Open set<\/strong>: Every point is an interior point (no boundary included)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Closed set<\/strong>: Contains all its boundary points<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">A disk without its edge is open. A disk with its edge is closed. Include only part of the edge? Neither open nor closed.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Connected Sets and Regions<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Connected<\/strong>: Can&#8217;t be split into separate pieces<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Region<\/strong>: Open, connected, and nonempty<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Limits at Boundary Points<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">When taking limits at boundary points, you only consider points inside both the [latex]\\delta[\/latex] disk and the domain. Some of the disk lies outside the domain\u2014that&#8217;s fine. Just ignore those points.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following limit:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,-2)}\\sqrt{29-x^{2}-y^{2}}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1667793733114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1667793733114\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y)\\to(5,-2)}\\sqrt{29-x^{2}-y^{2}}=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186151&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=zYJIT-NNVKM&amp;video_target=tpm-plugin-0y6dbuw0-zYJIT-NNVKM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.8_transcript.html\">transcript for \u201cCP 4.8\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Continuity<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A function of two variables is continuous at a point when there are no jumps, breaks, or holes at that point. The definition mirrors single-variable continuity.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Three Conditions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A function [latex]f(x, y)[\/latex] is continuous at [latex](a, b)[\/latex] if:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]f(a, b)[\/latex] exists<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f(x,y)[\/latex] exists<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f(x,y)=f(a,b)[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">All three must hold. The function must be defined, the limit must exist, and they must equal each other.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Checking Continuity<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Verify each condition:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Check the point is in the domain<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the limit using limit laws<\/li>\n<li class=\"whitespace-normal break-words\">Confirm the limit equals the function value<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Properties of Continuous Functions<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Sum<\/strong>: If [latex]f[\/latex] and [latex]g[\/latex] are continuous, so is [latex]f + g[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Product<\/strong>: If [latex]g(x)[\/latex] and [latex]h(y)[\/latex] are continuous, so is [latex]g(x)h(y)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Composition<\/strong>: If [latex]g(x, y)[\/latex] is continuous and [latex]f(z)[\/latex] is continuous, so is [latex]f(g(x, y))[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Common Continuous Functions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Polynomials, rational functions (where denominators aren&#8217;t zero), trig functions, exponentials, and logarithms are continuous on their domains. Compositions and combinations of these remain continuous.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Show that the function [latex]f\\,(x,y)=\\sqrt{26-2x^{2}-y^{2}}[\/latex]\u00a0is continuous at point [latex](2,-3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1667993733114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1667993733114\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"text-align: left;\">The domain of [latex]f[\/latex] contains the ordered pair [latex](2,-3)[\/latex] because [latex]f\\,(a,b)=f\\,(2,-3)=\\sqrt{16-2(2)^{2}-(-3)^{2}}=3[\/latex]<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=3[\/latex]<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=f\\,(a,b)=3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Show that the functions [latex]f\\,(x,\\ y)=2x^{2}y^{3}+3[\/latex]\u00a0and [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex] are continuous everywhere.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1687993733114\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1687993733114\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomials [latex]g\\,(x)=2x^{2}[\/latex] and [latex]h\\,(y)=y^{3}[\/latex]\u00a0are continuous at every real number; therefore, by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\u00a0Furthermore, any constant function is continuous everywhere, so [latex]g\\,(x,\\ y)=3[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Therefore, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Last, [latex]h\\,(x)=x^{4}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0so by the continuity of composite functions theorem [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186152&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k4SalFB3XEw&amp;video_target=tpm-plugin-310u6ibm-k4SalFB3XEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.10_transcript.html\">transcript for \u201cCP 4.10\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<h2>Functions of Three or More Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Limits and continuity extend naturally beyond two variables. The key is replacing [latex]\\delta[\/latex] disks with [latex]\\delta[\/latex] balls in higher dimensions.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The [latex]\\delta[\/latex] Ball<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">In three dimensions, a [latex]\\delta[\/latex] ball around [latex](x_0, y_0, z_0)[\/latex] includes all points within distance [latex]\\delta[\/latex]:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]{(x, y, z) \\mid \\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2} < \\delta}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">For four dimensions, add another squared term under the radical. The pattern continues for any number of dimensions.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Limits in Three Variables<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">The definition mirrors two variables. For [latex]\\displaystyle\\lim_{(x,y,z)\\to(a,b,c)}f(x,y,z)=L[\/latex], points in a [latex]\\delta[\/latex] ball around [latex](a,b,c)[\/latex] must give function values arbitrarily close to [latex]L[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Using Limit Laws<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">All limit laws from two variables work for three or more variables:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Sum, difference, product, quotient<\/li>\n<li class=\"whitespace-normal break-words\">Constant multiple, power, root<\/li>\n<li class=\"whitespace-normal break-words\">Identity laws<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Continuity<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A function is continuous at a point if the same three conditions hold: the function exists, the limit exists, and they&#8217;re equal. The concept is identical\u2014only the dimension changes.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1687993733119\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1687993733119\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}=2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-598","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"fresh_take","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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/598","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/598\/revisions"}],"predecessor-version":[{"id":1759,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/598\/revisions\/1759"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/598\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=598"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=598"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=598"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=598"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}