{"id":34007,"date":"2024-08-17T13:00:04","date_gmt":"2024-08-17T13:00:04","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34007"},"modified":"2025-08-01T07:00:50","modified_gmt":"2025-08-01T07:00:50","slug":"limites-laterales-definitiones-expositio-et-exercitia","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/limites-laterales-definitiones-expositio-et-exercitia\/","title":{"rendered":"Limites Laterales: Definitiones, Expositio et Exercitia"},"content":{"rendered":"<p><center><\/p>\n<h1>Limites Laterales: Definitiones, Expositio et Exercitia<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nHic articulus limites laterales et bilaterales in calculo explicat, ostendens quomodo et intuitive et formaliter definiantur atque applicentur. Exempla graphica et algebraica exhibentur, condiciones ad usum algebrae limitum in his casibus tractantur, et exercitia soluta includuntur ad intellectum confirmandum. Propositum est visionem claram et concisam horum notionum fundamentalium in studio calculi praebere.<\/em><\/p>\n<p><strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem discipulus poterit:<\/p>\n<ul style=\"text-align:left;\">\n<li><strong>Intelligere<\/strong> differentiam inter limites laterales et bilaterales.<\/li>\n<li><strong>Definire<\/strong> limites laterales ex dextra et sinistra formaliter.<\/li>\n<li><strong>Applicare<\/strong> definitionem limitum lateralium in problematis calculi.<\/li>\n<li><strong>Interpretari<\/strong> limites laterales et bilaterales graphice.<\/li>\n<li><strong>Agnoscere<\/strong> condiciones necessarias ad exsistentiam limitis bilateralis.<\/li>\n<li><strong>Adhibere<\/strong> algebram limitum in contextibus limitum lateralium.<\/li>\n<li><strong>Solvere<\/strong> exercitia calculi ad limites laterales pertinentia.<\/li>\n<\/ul>\n<p><strong>INDEX CONTENTORUM:<\/strong><br \/>\n<a href=\"#1\">Introductio<\/a><br \/>\n<a href=\"#2\">Notio Intuitiva Limitum Lateralium et Bilateralium<\/a><br \/>\n<a href=\"#3\">Definitio Formata Limitum Lateralium<\/a><br \/>\n<a href=\"#4\">Conditiones pro Algebra Limitum<\/a><br \/>\n<a href=\"#5\">Exercitia Proposita et Soluta<\/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>Introductio<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=138s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Limites laterales oriuntur cum de limitibus agitur qui solum ex sinistra vel dextra exsistere possunt,<\/span><\/strong><\/a> non autem ex utraque parte simul. Illi quos hactenus studuimus sunt quidem huius postremi generis: ut limen functionis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span> existat, necesse est ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> bene definita sit ex utraque parte puncti <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>; quodsi hoc non contingat, tum definitio limitis non valebit. Cum hae condiciones saepe occurrant, oportet modum inveniendi ad eas tractandas. Hoc per definitionem formalem solvitur.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Notio Intuitiva Limitum Lateralium et Bilateralium<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=279s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ut limen functionis exsistat<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span>, cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to x_0<\/span><\/span>, necesse est ut functio bene definita sit ex utraque parte puncti <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>. Si hoc evenit, de limite bilaterali agitur. Et si praeterea tale limen valorem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span> habeat, tunc recte scribere possumus<\/p>\n<p style=\"text-align: center;\"><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;\">Iam fingamus hanc functionem ita redefinire ut eius dominium tantummodo valores maiores quam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> includat. Si hoc facimus, animadvertimus limen iam non exsistere (quia erunt valores <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> quibus sensus deest); nihilominus, graphice adhuc dicere possumus quod, cum <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> adhuc tendit ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L<\/span><\/span>. Notio intuitiva quam hic producimus est limen laterale ex dextra, quod ita repraesentatur<\/p>\n<p style=\"text-align: center;\"><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;\">et similiter habebimus limen ex sinistra parte<\/p>\n<p style=\"text-align: center;\"><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;\">Denique, limen bilaterale exsistet dummodo limites laterales exsistant atque aequales sint<\/p>\n<p style=\"text-align: center;\"><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>Definitio Formata Limitum Lateralium<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=438s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ad limites laterales formaliter definiendos sufficit<\/span> <\/strong><\/a>parvam mutationem in definitione originali limitis adhibere.<\/p>\n<p style=\"text-align: center;\"><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;\">Pro limitibus lateralibus ex dextra definitio sic exprimetur:<\/p>\n<p style=\"text-align: center;\"><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;\">Pro limitibus lateralibus ex sinistra erit:<\/p>\n<p style=\"text-align: center;\"><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>Conditiones pro Algebra Limitum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=8lkvxEHsF6c&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Quod admirandum est de his definitionibus habendis, est quod ambae comprehenduntur<\/span><\/strong><\/a> simul in definitione limitis communis, quod grave est quia nos liberat ab onere iterum probandi omnes proprietates quas iam de limitibus bilateralibus ostendimus. Tota algebra limitum operabitur sicut in lectionibus prioribus vidimus, dummodo limites de eadem natura sint (ambo ex sinistra, vel ambo ex dextra, numquam confusi), eundem punctum spectent et in illo puncto exsistant.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercitia Proposita et Soluta<\/h2>\n<ol style=\"text-align: justify;\">\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;\">[SOLUTIO]<\/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>[SOLUTIO]<\/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>[SOLUTIO]<\/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>[SOLUTIO]<\/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>[SOLUTIO]<\/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;\">[SOLUTIO]<\/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;\">[SOLUTIO]<\/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;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limites Laterales: Definitiones, Expositio et Exercitia Summarium: Hic articulus limites laterales et bilaterales in calculo explicat, ostendens quomodo et intuitive et formaliter definiantur atque applicentur. Exempla graphica et algebraica exhibentur, condiciones ad usum algebrae limitum in his casibus tractantur, et exercitia soluta includuntur ad intellectum confirmandum. Propositum est visionem claram et concisam horum notionum fundamentalium [&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":1,"footnotes":""},"categories":[1328,1298],"tags":[],"class_list":["post-34007","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-calculus-differentialis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Limites Laterales: Definitiones, Expositio et Exercitia - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora expositionem accuratam limitum lateralium et bilateralium, cum definitionibus formalibus, exemplis practicis atque exercitiis solutis\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/limites-laterales-definitiones-expositio-et-exercitia\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Limites Laterales: Definitiones, Expositio et Exercitia\" \/>\n<meta property=\"og:description\" content=\"Explora expositionem accuratam limitum lateralium et bilateralium, cum definitionibus formalibus, exemplis practicis atque exercitiis solutis\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/limites-laterales-definitiones-expositio-et-exercitia\/\" \/>\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=\"2024-08-17T13:00:04+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-01T07:00:50+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=\"Limites Laterales: Definitiones, Expositio et Exercitia\" \/>\n<meta name=\"twitter:description\" content=\"Explora expositionem accuratam limitum lateralium et bilateralium, cum definitionibus formalibus, exemplis practicis atque exercitiis solutis\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Limites Laterales: Definitiones, Expositio et Exercitia\",\"datePublished\":\"2024-08-17T13:00:04+00:00\",\"dateModified\":\"2025-08-01T07:00:50+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/\"},\"wordCount\":787,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limiteslaterales.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limites-laterales-definitiones-expositio-et-exercitia\\\/\",\"name\":\"Limites Laterales: Definitiones, Expositio et Exercitia - 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