{"id":29461,"date":"2024-11-12T13:00:39","date_gmt":"2024-11-12T13:00:39","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29461"},"modified":"2024-11-13T07:40:49","modified_gmt":"2024-11-13T07:40:49","slug":"infinite-limits-and-divergence","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/infinite-limits-and-divergence\/","title":{"rendered":"Infinite Limits and Divergence"},"content":{"rendered":"<style>\np {\ntext-align: justify;}\n<\/style>\n<p><center><\/p>\n<h1>Infinite Limits and Divergence<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summary:<\/strong><br \/>\nIn this class, we will address infinite limits and the different types of divergence in limits, exploring fundamental concepts to understand how certain functions do not converge to a defined real value. We will review disparate lateral limits, infinitely oscillating functions, and situations where limits do not exist due to domain issues or unbounded growth.<br \/>\n<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this class, the student will be able to\n<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Define<\/strong> divergent limits and recognize when a limit is divergent.<\/li>\n<li><strong>Identify<\/strong> the different types of divergence in limits, such as disparate lateral limits and infinite limits.<\/li>\n<li><strong>Analyze<\/strong> situations where a function has domain issues and how this affects the existence of the limit.<\/li>\n<li><strong>Evaluate<\/strong> lateral limits to determine if they are disparate and the impact on limit convergence.<\/li>\n<li><strong>Calculate<\/strong> infinite limits and distinguish between limits that diverge toward positive and negative infinity.<\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n<u><strong>CONTENT INDEX<\/strong><\/u>:<br \/>\n<a href=\"#1\"><strong>When do we say a limit is divergent?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Types of Divergence in Limits<\/strong><\/a><br \/>\n<a href=\"#3\">Limits with Domain Issues<\/a><br \/>\n<a href=\"#4\">Disparate Lateral Limits<\/a><br \/>\n<a href=\"#5\">Limits of Infinitely Oscillating Functions<\/a><br \/>\n<a href=\"#6\">Infinite Limits<\/a><br \/>\n<a href=\"#7\">Infinite Limits at Infinity<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/SFBMSd0Q7Io\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>This time, we will not only review <strong>Infinite Limits<\/strong> but also <strong>divergent limits<\/strong> in general. Divergent limits tell us about how a function seems not to converge, and this can occur in many ways.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>When do we say a limit is divergent?<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=106s\" target=\"_blank\" rel=\"noopener\"><strong>We say a limit is divergent when it does not converge to any real value.<\/strong><\/a> This, which sounds obvious, can occur in different ways:<\/p>\n<ul>\n<li>When the lateral limits are different or non-existent, bilateral limits do not exist.<\/li>\n<li>If the function is not well-defined, grows without limit, or oscillates infinitely as it approaches the point where the limit is calculated, then the lateral limit cannot exist.<\/li>\n<\/ul>\n<p>This can apply, with its particularities, to both finite limits and limits at infinity, and depending on the case, we will have some type of divergence.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Types of Divergence in Limits<\/h2>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Limits with Domain Issues<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=210s\" target=\"_blank\" rel=\"noopener\"><strong>When we try to calculate a limit of the type <\/strong><\/a><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)<\/span> or <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x),<\/span> we expect at least that <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is well-defined for values close to <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span> or for some interval of the form <span class=\"katex-eq\" data-katex-display=\"false\">[a,+\\infty[, <\/span> respectively. If such a thing does not happen, then neither definition of limits could even make sense; the function cannot \u00abtend\u00bb to some value if it approaches from where it is not even defined. In such cases, we simply write that the limit does not exist: <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to x_0}f(x)=\\cancel{\\exists}<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to +\\infty}f(x)=\\cancel{\\exists},<\/span> as appropriate. Similarly, this applies to lateral limits, and nothing more can be said about this type of situation.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Disparate Lateral Limits<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=180s\" target=\"_blank\" rel=\"noopener\"><strong>Consider a function of the type<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = x\/|x|<\/span> and calculate the limit when <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>. The first thing we notice will be that<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} f(x) = 1<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} f(x) = -1<\/span>\n<p>In this case, we note that while the lateral limits exist, they are different. When this occurs, we simply say that the (bilateral) limit does not converge, and therefore:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} f(x) = \\cancel{\\exists}<\/span>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Limits of Infinitely Oscillating Functions<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=415s\" target=\"_blank\" rel=\"noopener\"><strong>There is also the case where functions, instead of approaching a certain value,<\/strong><\/a> begin to oscillate within a certain range. An example of this would be a function of the type <span class=\"katex-eq\" data-katex-display=\"false\">f(x)= \\sin(1\/x)<\/span>. If we observe what happens with this function when <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, we will see that it oscillates infinitely.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-vrdcIAmLV50\/YGoC903Q62I\/AAAAAAAAE1M\/6AM7D-RM9lAy9ZrG105MbfCN8ltu6J09ACLcBGAsYHQ\/s0\/sin1sobrex.PNG\" alt=\"f(x) = sin(1\/x)\" class=\"alignnone size-full lazyload\" width=\"856\" height=\"442\" \/><\/noscript><\/p>\n<p>When similar things happen, we say that the limit simply does not exist.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Infinite Limits<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=658s\" target=\"_blank\" rel=\"noopener\"><strong>Let\u2019s see what happens with the function<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x.<\/span> The first thing we will see is that when <span class=\"katex-eq\" data-katex-display=\"false\">x\\to 0<\/span>, the value of <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> grows without limit, but the way it will do so will depend on where the limit is calculated from. Intuitively, we will write<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^+} \\dfrac{1}{x} = +\\infty<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0^-} \\dfrac{1}{x} = -\\infty<\/span>\n<p>With this notation, we are not saying that the limit exists in some way; instead, we are indicating how this limit does not exist. Unlike previous cases, where the limit does not exist and does not converge to a specific value; in this case, it diverges because its size goes beyond any real number.<\/p>\n<p>What we just reviewed can be formalized through the following definitions:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow M \\lt f(x) )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = +\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = +\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = +\\infty \\right)<\/span>\n<p>And similarly:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^+}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 \\lt x \\lt x_0 + \\delta \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0^-}f(x) = -\\infty := \\left(\\forall m \\in \\mathbb{R}\\right)\\left( \\exists \\delta \\gt 0 \\right) ( x_0 - \\delta \\lt x \\lt x_0 \\rightarrow f(x) \\lt m )<\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = -\\infty := \\left(\\lim_{x\\to x_0^+}f(x) = -\\infty \\right) \\wedge \\left(\\lim_{x\\to x_0^-}f(x) = -\\infty \\right)<\/span>\n<p>Sometimes, we also talk about limits that tend to infinity (without sign)<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to x_0}f(x) = \\infty := \\lim_{x\\to x_0}|f(x)| = +\\infty <\/span>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Infinite Limits at Infinity<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=SFBMSd0Q7Io&amp;t=1147s\" target=\"_blank\" rel=\"noopener\"><strong>Similarly to the limits reviewed above,<\/strong><\/a> it is possible to define infinite limits at infinity. For example:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty}f(x) = +\\infty := \\left(\\forall M \\in \\mathbb{R}\\right)\\left( \\exists N \\in\\mathbb{R} \\right) ( N\\lt x \\rightarrow M \\lt f(x) )<\/span>\n<p>And with this, we have seen all the ways in which functions\u2019 limits can diverge.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Infinite Limits and Divergence Summary: In this class, we will address infinite limits and the different types of divergence in limits, exploring fundamental concepts to understand how certain functions do not converge to a defined real value. We will review disparate lateral limits, infinitely oscillating functions, and situations where limits do not exist due to [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29458,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":12,"footnotes":""},"categories":[854,567],"tags":[],"class_list":["post-29461","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-differential-calculus","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Infinite Limits and Divergence - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Here, we will review how infinite limits are defined and what function divergence means. 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