{"id":27975,"date":"2024-08-17T13:00:26","date_gmt":"2024-08-17T13:00:26","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27975"},"modified":"2024-08-16T01:00:44","modified_gmt":"2024-08-16T01:00:44","slug":"lateral-limits-definitions-development-and-exercises","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/","title":{"rendered":"Lateral Limits: Definitions, Development, and Exercises"},"content":{"rendered":"<p><center><\/p>\n<h1>One-Sided Limits: Definitions, Development, and Exercises<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nThis post explains one-sided and bilateral limits in calculus, showing how they are defined and applied both intuitively and formally. Graphical and algebraic examples are presented, the conditions for using limit algebra in these cases are discussed, and solved exercises are included to reinforce understanding. The goal is to provide a clear and concise view of these fundamental concepts in the study of calculus.<\/em><\/p>\n<p><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this lesson, the student will be able to:<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the difference between one-sided and bilateral limits.<\/li>\n<li><strong>Formally define<\/strong> one-sided limits from the right and left.<\/li>\n<li><strong>Apply<\/strong> the definition of one-sided limits to calculus problems.<\/li>\n<li><strong>Graphically interpret<\/strong> one-sided and bilateral limits.<\/li>\n<li><strong>Recognize<\/strong> the necessary conditions for a bilateral limit to exist.<\/li>\n<li><strong>Use<\/strong> limit algebra in the context of one-sided limits.<\/li>\n<li><strong>Solve<\/strong> calculus exercises related to one-sided limits.<\/li>\n<\/ul>\n<p><strong>TABLE OF CONTENTS:<\/strong><br \/>\n<a href=\"#1\">Introduction<\/a><br \/>\n<a href=\"#2\">Intuitive Idea of One-Sided and Bilateral Limits<\/a><br \/>\n<a href=\"#3\">Formal Definition of One-Sided Limits<\/a><br \/>\n<a href=\"#4\">Conditions for Limit Algebra<\/a><br \/>\n<a href=\"#5\">Proposed and Solved Exercises<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/8lkvxEHsF6c\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=138s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">One-sided limits arise when we encounter limits that could only exist from the left or right,<\/span><\/strong><\/a> but not from both sides. The ones we have studied so far are precisely of this last kind: for the limit of the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span> to exist, it is necessary that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> be well-defined on both sides of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>; if this does not happen, then the definition of the limit will not work. Since situations where such limits occur are frequent, it is necessary to find a way to deal with them. This is resolved through a formal definition.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Intuitive Idea of One-Sided and Bilateral Limits<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=279s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">For the limit of a function to exist<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span>, when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, it is necessary that the function be well-defined on both sides of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. If this occurs, then we talk about a bilateral limit. And if such a limit also results in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>, then there would be no problem writing<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now, imagine that we redefine this function in such a way that its domain only includes values greater than <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. If we do this, we will notice that the limit has ceased to exist (because there will be values of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> for which it will not make sense); however, graphically we could still say that when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> still tends to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>. The intuitive idea we are producing here is that of a right-hand limit, which is what we would represent through the notation<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^+}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">and similarly, we will have the left-hand limit<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Finally, the bilateral limit will exist whenever the one-sided limits exist and are equal<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0^-}f(x) = \\lim_{x\\to x_0}f(x) = \\lim_{x\\to x_0^-}f(x)<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Formal Definition of One-Sided Limits<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=438s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">To formally define one-sided limits, it is sufficient<\/span> <\/strong><\/a>to apply a small modification to the original limit definition.<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} f(x) = L := \\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0\\lt|x-x_0|\\lt\\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">For right-hand limits, the definition is as follows:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\lt x \\lt x_0 + \\delta <\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">For left-hand limits, it will be:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall \\epsilon\\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\exist \\delta \\gt 0 \\right)<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\right.<\/span><\/span> <span style=\"background-color: #88ff88;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 - \\delta \\lt x \\lt x_0<\/span><\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\rightarrow |f(x)-L|\\lt\\epsilon \\right)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Conditions for Limit Algebra<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The interesting thing about having these definitions is that they are both contained<\/span><\/strong><\/a> within the usual limit definition, and this is important because it frees us from having to re-prove all the properties we already proved for bilateral limits. All the limit algebra will work just as we have seen in previous classes, as long as the limits involved are of the same nature (both from the left, or both from the right, never mixed), are directed to the same point, and exist at that point.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Proposed and Solved Exercises<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to {\\frac{1}{2}}^- } \\sqrt{\\dfrac{x+2}{x+1}} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=994s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\sqrt{\\dfrac{x-1}{x+2}} <\/span><\/span><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 2^+} \\left(\\dfrac{x}{x+1} \\right) \\left(\\dfrac{2x+5}{x^2+x} \\right)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1058s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-} \\left(\\dfrac{1}{x+1} \\right) \\left(\\dfrac{x+6}{x} \\right) \\left(\\dfrac{3-x}{x} \\right) <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1102s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^+ } \\dfrac{\\sqrt{h^2 + 4h +5} - \\sqrt{5}}{h} <\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1168s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{h\\to 0^-} \\dfrac{\\sqrt{6} - \\sqrt{5h^2 + 11h +6}}{h}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1276s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^+} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -2^-} (x+3)\\dfrac{|x+2|}{x+2}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1378s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li>\n<table>\n<tbody>\n<tr style=\"text-align: justify;\">\n<td>a. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^+} \\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>b. <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 1^-}\\dfrac{\\sqrt{2x}(x-1)}{|x-1|}\n\n<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=1655s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>One-Sided Limits: Definitions, Development, and Exercises Summary: This post explains one-sided and bilateral limits in calculus, showing how they are defined and applied both intuitively and formally. Graphical and algebraic examples are presented, the conditions for using limit algebra in these cases are discussed, and solved exercises are included to reinforce understanding. The goal is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27972,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":128,"footnotes":""},"categories":[854,567],"tags":[],"class_list":["post-27975","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>Lateral Limits: Definitions, Development, and Exercises - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Lateral Limits: Definitions, Development, and Exercises\" \/>\n<meta property=\"og:description\" content=\"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2024-08-17T13:00:26+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Lateral Limits: Definitions, Development, and Exercises\" \/>\n<meta name=\"twitter:description\" content=\"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Lateral Limits: Definitions, Development, and Exercises\",\"datePublished\":\"2024-08-17T13:00:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/\"},\"wordCount\":912,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"articleSection\":[\"Differential Calculus\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/\",\"name\":\"Lateral Limits: Definitions, Development, and Exercises - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"datePublished\":\"2024-08-17T13:00:26+00:00\",\"description\":\"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"width\":1024,\"height\":347},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/lateral-limits-definitions-development-and-exercises\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Lateral Limits: Definitions, Development, and Exercises\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Lateral Limits: Definitions, Development, and Exercises - toposuranos.com\/material","description":"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/","og_locale":"es_ES","og_type":"article","og_title":"Lateral Limits: Definitions, Development, and Exercises","og_description":"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises","og_url":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2024-08-17T13:00:26+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Lateral Limits: Definitions, Development, and Exercises","twitter_description":"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Lateral Limits: Definitions, Development, and Exercises","datePublished":"2024-08-17T13:00:26+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/"},"wordCount":912,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","articleSection":["Differential Calculus","Mathematics"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/","url":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/","name":"Lateral Limits: Definitions, Development, and Exercises - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","datePublished":"2024-08-17T13:00:26+00:00","description":"Explore a detailed explanation of lateral and bilateral limits with formal definitions, practical examples, and solved exercises","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limiteslaterales.jpg","width":1024,"height":347},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/en\/lateral-limits-definitions-development-and-exercises\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Lateral Limits: Definitions, Development, and Exercises"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27975","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=27975"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27975\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/27972"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=27975"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=27975"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=27975"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}