{"id":27911,"date":"2024-08-16T13:00:51","date_gmt":"2024-08-16T13:00:51","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27911"},"modified":"2024-08-18T06:54:59","modified_gmt":"2024-08-18T06:54:59","slug":"el-teorema-del-sandwich-para-el-calculo-de-limites","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/es\/el-teorema-del-sandwich-para-el-calculo-de-limites\/","title":{"rendered":"El Teorema del Sandwich para el C\u00e1lculo de L\u00edmites"},"content":{"rendered":"<p><center><\/p>\n<h1>El Teorema del Sandwich para el C\u00e1lculo de L\u00edmites<\/h1>\n<p><em><strong>Resumen:<\/strong><br \/>\nEsta clase presenta el Teorema del Sandwich, una herramienta clave en c\u00e1lculo para evaluar l\u00edmites dif\u00edciles utilizando funciones m\u00e1s sencillas que acotan por arriba y por abajo. Se ofrece una explicaci\u00f3n gr\u00e1fica y una demostraci\u00f3n formal, seguida de ejemplos pr\u00e1cticos. El objetivo es que los estudiantes comprendan c\u00f3mo aplicar este teorema para calcular l\u00edmites de manera m\u00e1s eficiente.<\/em><\/p>\n<p><strong>Objetivos de Aprendizaje:<\/strong><br \/>\nAl completar esta clase, el estudiante ser\u00e1 capaz de<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Comprender<\/strong> la utilidad del Teorema del Sandwich en el c\u00e1lculo de l\u00edmites.<\/li>\n<li><strong>Identificar<\/strong> funciones que pueden acotar una funci\u00f3n objetivo para aplicar el teorema.<\/li>\n<li><strong>Aplicar<\/strong> el Teorema del Sandwich para calcular l\u00edmites dif\u00edciles.<\/li>\n<li><strong>Visualizar<\/strong> gr\u00e1ficamente el concepto del Teorema del Sandwich.<\/li>\n<li><strong>Demostrar<\/strong> el Teorema del Sandwich de manera formal.<\/li>\n<\/ul>\n<p><strong><u>\u00cdNDICE DE CONTENIDOS<\/u>:<\/strong><br \/>\n<a href=\"#1\">Introducci\u00f3n<\/a><br \/>\n<a href=\"#2\">Idea Gr\u00e1fica del Teorema del Sandwich<\/a><br \/>\n<a href=\"#3\">Demostraci\u00f3n del Teorema del Sandwich<\/a><br \/>\n<a href=\"#4\">Ejemplos<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/24G_qlEwL9M\" 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>Introducci\u00f3n<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=158s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La utilidad del teorema del Sandwich radica en la facilidad que otorga al c\u00e1lculo de algunos l\u00edmites dificiles<\/span><\/strong><\/a> a trav\u00e9s de otros mas sencillos. La raz\u00f3n del nombre es que, en lugar de calcular directamente el l\u00edmite de una funci\u00f3n cuando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, se usa otro par de funciones, una acotando por arriba y la otra por abajo, y cuyo l\u00edmite en <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> coincide y es f\u00e1cil de obtener. Como la funci\u00f3n original est\u00e1 siempre entre las dos, esta queda como \u00abel queso entre entre los dos trozos de pan\u00bb.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Idea Gr\u00e1fica del Teorema del Sandwich<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=206s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La idea que sintetiza el teorema en realidad es bastante sencilla.<\/span> <\/strong><\/a>Supongamos que queremos calcular cierto l\u00edmite dificil<\/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)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Lo que normalmente se hace es tomar todo nuestro conocimiento del \u00e1lgebra de funciones para intentar <strong>simplificarlo hasta el punto en que lo podamos evaluar<\/strong>. Sin embargo, en ocasiones un enfoque distinto es mucho m\u00e1s eficiente. Supongamos que tenemos un intervalo cerrado <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I<\/span><\/span> tal que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0 \\in I<\/span><\/span> y adem\u00e1s existen otras dos funciones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m(x)<\/span><\/span> y <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M(x)<\/span><\/span> que satisfacen la relaci\u00f3n<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in I)(m(x)\\leq f(x) \\leq M(x) )<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Y que adem\u00e1s<\/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} m(x) = \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Entonces se cumplir\u00e1 que<\/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;\">Esto es lo que podemos ver en la siguiente im\u00e1gen.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"teorema del sandwich\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-gjBfVdaLj-k\/YGDXVUQBqDI\/AAAAAAAAEvg\/d2sNJdweVaoB64O5e2qfBxjZGIyIyGOxgCLcBGAsYHQ\/s0\/teorema%2Bdel%2Bsandwich.PNG\" alt=\"teorema del sandwich\" class=\"alignnone size-full lazyload\" width=\"513\" height=\"407\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Demostraci\u00f3n del Teorema del Sandwich<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=404s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Para demostrar el teorema del Sandwich<\/span><\/strong><\/a>, seguiremos el siguiente razonamiento:<\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0\\in I<\/span><\/span><\/span>; <strong>Premisa<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} m(x) = L<\/span><\/span><\/span> ; <strong>Premisa<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_1 \\gt 0) (|x-x_0|\\lt \\delta_1 \\rightarrow |m(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} M(x) = L<\/span><\/span><\/span> ; <strong>Premisa<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta_2 \\gt 0) (|x-x_0|\\lt \\delta_2 \\rightarrow |M(x) -L| \\lt \\epsilon )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td><span style=\"background-color: #90ff90;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) \\leq f(x) \\leq M(x) )<\/span><\/span><\/span>; <strong>Premisa<\/strong><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x \\in I)(m(x) - L \\leq f(x) - L \\leq M(x) - L )<\/span><\/span>; De(4)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|m(x) -L|\\lt \\epsilon) \\rightarrow (-\\epsilon \\lt m(x) - L \\lt \\epsilon)<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(|M(x) -L|\\lt \\epsilon ) \\rightarrow (-\\epsilon \\lt M(x) - L \\lt \\epsilon) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( |M(x) -L| \\lt \\epsilon \\wedge |m(x) -L| \\lt \\epsilon ) )<\/span><\/span>; de (2,3)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow ( - \\epsilon \\lt f(x) - L \\lt \\epsilon ) )<\/span><\/span>; de (1,5,6,7,8)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0)(\\exists \\delta \\gt 0) (|x-x_0|\\lt \\delta=\\min\\{\\delta_1,\\delta_2\\} \\rightarrow |f(x) - L| \\lt \\epsilon ) )<\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}f(x) = L\\;\\blacksquare<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Ejemplos<\/h2>\n<p style=\"text-align: justify; color: #000000;\">Utilizando el Teorema del Sandwich, podemos calcular el l\u00edmite de funciones incluso cuando ni siquiera tenemos su expresi\u00f3n algebr\u00e1ica de forma explicita. A continuaci\u00f3n un par de ejemplos de esto:<\/p>\n<p style=\"text-align: justify; color: #000000;\">Un ejemplo de esto se da en las siguiente situacion:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{5-2x^2}\\leq f(x) \\leq \\sqrt{5-x^2}<\/span><\/span>, cuando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-1\\leq x\\leq 1<\/span><\/span>. \u00bfCu\u00e1l es el valor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}f(x)<\/span><\/span>? <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1082s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUCI\u00d3N]<\/span><\/strong><\/a><\/li>\n<\/ul>\n<p style=\"text-align: justify; color: #000000;\">Otro uso pr\u00e1ctico del teorema del Sandwich se da cuando el limite en si mismo no es evidente respecto a otros m\u00e1s sencillos que le acotan por arriba y por abajo, como es lo que se obtiene al calcular el siguiente caso:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li>Calcular: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to 0}\\dfrac{\\sin(x)}{x}<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=24G_qlEwL9M&amp;t=1157s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUCI\u00d3N]<\/span><\/strong><\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El Teorema del Sandwich para el C\u00e1lculo de L\u00edmites Resumen: Esta clase presenta el Teorema del Sandwich, una herramienta clave en c\u00e1lculo para evaluar l\u00edmites dif\u00edciles utilizando funciones m\u00e1s sencillas que acotan por arriba y por abajo. Se ofrece una explicaci\u00f3n gr\u00e1fica y una demostraci\u00f3n formal, seguida de ejemplos pr\u00e1cticos. El objetivo es que los [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27930,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":201,"footnotes":""},"categories":[852,563],"tags":[],"class_list":["post-27911","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculo-diferencial","category-matematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>El Teorema del Sandwich para el C\u00e1lculo de L\u00edmites - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprende a calcular l\u00edmites utilizando el Teorema del Sandwich. 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