{"id":29558,"date":"2024-11-18T17:33:10","date_gmt":"2024-11-18T17:33:10","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29558"},"modified":"2024-11-18T17:33:10","modified_gmt":"2024-11-18T17:33:10","slug":"limit-and-continuity","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/limit-and-continuity\/","title":{"rendered":"Limit and Continuity"},"content":{"rendered":"<style>\np {\ntext-align:justify;\n}\n<\/style>\n<p><center><\/p>\n<h1>Limit and Continuity<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Summary:<\/strong><\/em><br \/>\nThis class addresses the relationship between the limit and the continuity of a function, starting with an intuitive and formal explanation of the term. Continuity at a point and in a set is explored, detailing the necessary conditions for a function to be continuous. The algebraic properties of continuous functions are also presented, including sums, products, quotients, and function composition. Finally, practical examples are solved to analyze the continuity of different functions and calculate limits, relating them to the concept of continuity.\n<\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><br \/>\nAt the end of this class, the student will be able to:<\/p>\n<ol>\n<li><strong>Analyze<\/strong> the continuity of a function at a point by verifying the defined conditions.<\/li>\n<li><strong>Evaluate<\/strong> whether a function is continuous on a given set using the formal definition of continuity.<\/li>\n<li><strong>Demonstrate<\/strong> the continuity of a function based on its algebraic properties.<\/li>\n<li><strong>Classify<\/strong> points of discontinuity into different types according to the conditions that are not met.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>CONTENT INDEX<\/u>:<\/strong><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">Continuity at a Point<\/a><br \/>\n<a href=\"#3\">Continuity in a Set<\/a><br \/>\n<a href=\"#4\">Properties of Continuous Functions<\/a><br \/>\n<a href=\"#5\">Exercises<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/4kehfJcyPRg\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=109s\" target=\"_blank\" rel=\"noopener\"><strong>Intuitively speaking, continuity<\/strong><\/a> of a function is understood as the property that allows its graph to be drawn \u00abwithout lifting the pencil.\u00bb This notion, which is quite simple to understand on its own, is not sufficient, however, if mathematical rigor is desired. A minimum of formalities is required. Formally speaking, the continuity of functions is closely linked to the concept of limit, and this is what we will study in detail.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Continuity at a Point<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=152s\" target=\"_blank\" rel=\"noopener\"><strong>A function <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is continuous at a point<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">x=x_0<\/span> if and only if the following conditions are met:<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is defined at <span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/li>\n<li>The limit <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x)<\/span> exists<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = f(x_0)<\/span><\/li>\n<\/ul>\n<p >All of this can be mathematically synthesized through the expression<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p >This is precisely the mathematical definition of a limit, where the value \u00abL\u00bb has been replaced by \u00abf(<span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span>)\u00bb<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Continuity in a Set<\/h3>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=462s\" target=\"_blank\" rel=\"noopener\"><strong>The definition of pointwise continuity<\/strong><\/a> can be extended to continuity for all points within a certain set. We say that <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> is continuous within a set <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> if it is continuous at all points <span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I.<\/span> This can be mathematically synthesized through the following expression.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x_0 \\in I\\subseteq Dom(f) \\right)\\left(\\forall \\epsilon \\gt 0 \\right) \\left(\\exists \\delta \\gt 0 \\right) \\left(0\\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - f(x_0)|\\lt \\epsilon\\right)<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Properties of Continuous Functions<\/h3>\n<h4>Algebra of Continuous Functions<\/h4>\n<p ><a href=\"https:\/\/www.youtube.com\/watch?v=4kehfJcyPRg&amp;t=575s\" target=\"_blank\" rel=\"noopener\"><strong>If <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> are continuous functions<\/strong><\/a> on <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)\\cap Dom(g)<\/span> then we will have:<\/p>\n<ul >\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\pm g<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\\cdot g<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li>If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\in\\mathbb{R}<\/span>, then <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha f<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">f\/g<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> as long as <span class=\"katex-eq\" data-katex-display=\"false\">g\\neq 0<\/span><\/li>\n<li>If <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta\\in\\mathbb{R}<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">\\beta\\neq 0<\/span>, then <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>, as long as <span class=\"katex-eq\" data-katex-display=\"false\">f^{\\alpha\/\\beta}<\/span> is well defined on <span class=\"katex-eq\" data-katex-display=\"false\">I.<\/span><\/li>\n<\/ul>\n<p >All these properties, which seem difficult to prove from the formal definition of continuity, are actually very easy because they are practically analogous to the algebra of limits that <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" rel=\"noopener\" target=\"_blank\">we have already proven<\/a>, so we are exempt from doing them.<\/p>\n<h5>Composition of Continuous Functions<\/h5>\n<p >If <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I\\subseteq Dom(f)<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">f(I)<\/span>, then the composition <span class=\"katex-eq\" data-katex-display=\"false\">(g\\circ f)(x) = g(f(x))<\/span> is continuous on <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span>. This, as is already inferred, is a direct consequence of the laws of composition for the limits of functions.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercises<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/R91vswfQT_Q\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<ol >\n<li>Analyze the continuity of the following functions<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{1}{x-2} - 3x<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=33s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+1}{x^2-4x+3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=97s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=|x-1| + \\sin(x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=215s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\cos(x)}{x}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=csc(2x)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=564s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x\\tan(x)}{x^2 + 1}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=666s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>g.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\sqrt{2x + 3}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=784s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>h.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{x+2}{\\cos(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=917s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>i.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y=\\dfrac{\\sqrt{x^4 + 1}}{1+\\sin^2(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=952s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Calculate the following limits. Are the functions continuous at the points they approach?<br \/>\n<table>\n<tbody>\n<tr>\n<td width=\"30\">a.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to\\pi} \\sin(x -\\sin(x))<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1033s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>b.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\sin\\left(\\dfrac{\\pi}{2}\\cos(\\tan(x)) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1092s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>c.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 1} \\sec\\left(x\\sec^2(x) - \\tan^2(x) -1 \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1162s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>d.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\tan\\left(\\dfrac{\\pi}{4}\\cos\\left(\\sin\\left(x^{1\/3}\\right)\\right) \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1274s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>e.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to 0} \\cos\\left(\\dfrac{\\pi}{\\sqrt{19 - 3\\sec{2x}}} \\right)<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1424s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td>f.<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to \\pi\/6} \\sqrt{\\csc^2(x) + 5\\sqrt{3}\\tan(x)}<\/span><\/td>\n<td><a href=\"https:\/\/www.youtube.com\/watch?v=R91vswfQT_Q&amp;t=1505s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limit and Continuity Summary: This class addresses the relationship between the limit and the continuity of a function, starting with an intuitive and formal explanation of the term. Continuity at a point and in a set is explored, detailing the necessary conditions for a function to be continuous. The algebraic properties of continuous functions are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29557,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[854,567],"tags":[],"class_list":["post-29558","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Limit and Continuity - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"What is continuity and how is it related to the limit? 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