{"id":33311,"date":"2024-08-11T13:00:29","date_gmt":"2024-08-11T13:00:29","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=33311"},"modified":"2025-07-21T21:00:41","modified_gmt":"2025-07-21T21:00:41","slug":"limes-functionum-unius-variabilis-realis","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/limes-functionum-unius-variabilis-realis\/","title":{"rendered":"Limes Functionum Unius Variabilis Realis"},"content":{"rendered":"<p><center><\/p>\n<h1>Limes Functionum Variabilis Realis<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center\"><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione perscrutatur definicio formalis limi\u00adnis functionum variabilis realis, ex qua demonstrantur proprietates praecipuae quae ad algebram limitum ducunt.<\/br><\/em><\/p>\n<p style=\"text-align:center\"><em><strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem discipulus poterit:<\/p>\n<ul>\n<li><strong>Meminisse<\/strong> definitionem limi\u00adnis functionum variabilis realis.<\/li>\n<li><strong>Demonstr\u0101re<\/strong> proprietates quae ad algebram limitum ducunt per deductiones <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon-\\delta<\/span><\/span>.<\/li>\n<li><strong>Comput\u0101re<\/strong> limites functionum variabilis realis utens algebra limitum eiusque proprietatibus.<\/li>\n<\/ul>\n<p><\/em><\/p>\n<p><center><br \/>\n<strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\"><strong>Introductio<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Notio Intuitiva Limiti Functionis ex Aspectu Graphico<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>Definitio Formalis Limiti<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>Proprietates Limitum<\/strong><\/a><br \/>\n<a href=\"#5\">Si limes existit, tunc est unicus<\/a><br \/>\n<a href=\"#6\">Algebra Limitum<\/a><br \/>\n<a href=\"#7\">Computatio Limitum Simplicium<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/FEPfoAfPsFY\" 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>Introductio<\/h2>\n<p style=\"text-align: justify;\">\n  <a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=240s\" rel=\"noopener\" target=\"_blank\"><br \/>\n    <strong>Quid interest inter studium algebrae et geometriae respectu studii calculi?<\/strong><br \/>\n  <\/a><br \/>\n  Responsum ad hanc quaestionem nobis datur per notionem limi\u00adnis. Hoc in articulo ergo studetur limes eiusque definitio.\n<\/p>\n<p style=\"text-align: justify;\">\n  Verbum \u201climes\u201d plerumque connectitur cum quadam specie finis, sicut finis intervalli cuius termini sunt a, b (quaecumque eorum natura sit).\n<\/p>\n<p style=\"text-align: center;\">\n  <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b[\\;\\; ;\\;\\; ]a,b]\\;\\; ; \\;\\; ]a,b[\\;\\; ; [a,b]<\/span><\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n  Aut sicut praesens, de quo dici potest esse finis inter praeteritum et futurum. Similiter ac modo quodam, notio limi\u00adtis inducit intellegentiam mathematicam huius notitiae intuitivae access\u016bs asymptotici ad punctum quendam.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Notio Intuitiva Limiti Functionis ex Aspectu Graphico<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=314s\" rel=\"noopener\" target=\"_blank\"><strong>Ad ideam limi\u00adtis percipiendam, oportet initium facere repraesentatione graphica<\/strong><\/a> functionis et quaerere quid eveniat cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> dum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> quantumvis appropinquat.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"limes functionis\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-rMjBnCIK8Ts\/YGDfXYswS4I\/AAAAAAAAEwA\/1GY0wy3JkXk99kveDTp1SltJOTAITgN3wCLcBGAsYHQ\/s0\/limite.PNG\" alt=\"limes functionis\" class=\"alignnone size-full lazyload\" width=\"692\" height=\"565\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> est prope <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, tunc exsistet intervallum apertum radii <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> et centrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span>, ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> in eo contineatur. Hoc tripliciter exprimi potest:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|x-x_0|\\lt \\delta<\/span><\/span>,<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]x_0 - \\delta , x_0 + \\delta[ <\/span><\/span>,<\/p>\n<p style=\"text-align: center;\">vel <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathcal{B}(x_0,\\delta)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\"><em>In nostro contextu, hae sunt tres formae ad idem exprimendum; attamen illa postrema, quae legitur ut \u00ab<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> in sphaera aperta centro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> et radio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> contentum esse\u00bb, aptior esset in <strong>cursu topologiae,<\/strong> ubi haec \u00abnotio propinquitatis\u00bb multo altius exploraretur.<\/em><\/p>\n<p style=\"text-align: justify;\">Si hoc fit, tunc observabimus existere aliud intervallum apertum centro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> et radio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> tale ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> in eo contineatur, id est: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon<\/span><\/span>.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"limes functionis\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-26xU-o1y-Eg\/YGDfXYgOp2I\/AAAAAAAAEwE\/FGMGEQdvRzg_OvnUqKolJ9v51xUVF4O7QCLcBGAsYHQ\/s0\/limite2.PNG\" alt=\"limes functionis\" class=\"alignnone size-full lazyload\" width=\"625\" height=\"549\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Hinc emergit idea fundamentalis notioni mathematicae limi\u00adtis, ex eo quod hic exsistet cum: si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span>, tum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)-l|\\lt \\epsilon<\/span><\/span>; atque hoc valor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span> erit limes functionis cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0<\/span><\/span> quantumlibet appropinquat.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Definitio Formalis Limiti<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=689s\" rel=\"noopener\" target=\"_blank\"><strong>Ex notione intuitiva et graphica supra proposita incipere possumus ad definitionem formalem limi\u00adtis explicandam.<\/strong><\/a> Dicimus limitem exsistere cum, quidquid sit hoc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span><\/span> (id est, distantia inter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">l<\/span><\/span>), semper exsistet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta<\/span><\/span> tale ut, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\lt|x-x_0|\\lt \\delta<\/span><\/span> satisfit, tunc etiam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - l|\\lt \\epsilon.<\/span><\/span> Hoc principium, quod initio difficile est comprehendere et multis studentibus calculi lacrimas affert per orbem terrarum, compendiari potest hac formula:<\/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)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - l|\\lt \\epsilon\\right)<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Proprietates Limitum<\/h2>\n<p style=\"text-align: justify;\">Momentum definitionis formalis limi\u00adtis est hoc: ex ea nunc possumus proprietates demonstrare, tum illas quae intuitive videntur verae tum etiam alias minus intuitivas.<\/p>\n<p style=\"text-align: justify;\">Antequam pergamus, quamquam non est absolute necessarium, valde commendatur ut recognoscas aliquos conceptus <a href=\"http:\/\/toposuranos.com\/material\/es\/category\/matematica\/logica-matematica\/logica-proposicional\/\" rel=\"noopener\" target=\"_blank\"><strong>logicae mathematicae<\/strong><\/a>, ut facilius intellegas demonstrationes quae infra sequuntur.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Si limes exsistit, tunc est unicus<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=904s\" rel=\"noopener\" target=\"_blank\"><strong>Ad hanc proprietatem demonstrandam, adhibebimus rationem per absurdum.<\/strong><\/a> Incipiemus definientes sequentem collectionem praemissarum:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}= \\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\}<\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Ex hoc construere possumus sequentem probationem formalem:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; <strong>Praesumptio<\/strong><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L|\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L^\\prime <\/span><\/span>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow |f(x) - L^\\prime |\\lt \\epsilon\\right) <\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\" text-align: left;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash L \\neq L^\\prime <\/span><\/span>; Praesumptio<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span><br \/>\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span>&#8211;<strong>Intro<\/strong>(1,2)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\forall \\epsilon \\gt 0\\right)\\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span><br \/>\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\epsilon \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\epsilon\\right) \\right] \\right. <\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; <strong>Monotonia<\/strong>(4)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\epsilon = \\frac{L - L^\\prime}{2}\\gt 0 <\/span><\/span>; Quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">L \\lt L^\\prime <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\left(\\exists \\delta\\gt 0\\right) \\left(0 \\lt|x-x_0|\\lt\\delta \\rightarrow\\right.<\/span><\/span><br \/>\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\left[ \\left( |f(x) - L |\\lt \\frac{L - L^\\prime}{2} \\right) \\wedge \\left( |f(x) - L^\\prime |\\lt \\frac{L - L^\\prime}{2}\\right) \\right] \\right. <\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">)<\/span><\/span>; Utendo(5,6)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2 |f(x) - L |\\lt L - L^\\prime )<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( 2|f(x) - L^\\prime |\\lt L - L^\\prime)<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2 (f(x) - L )\\lt L - L^\\prime )<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2(f(x) - L^\\prime )\\lt L - L^\\prime)<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L \\lt L - L^\\prime )<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + L^\\prime \\lt 2f(x) - 2L^\\prime \\lt L - L^\\prime)<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash (\\exists \\delta\\gt 0) (0 \\lt|x-x_0|\\lt\\delta \\rightarrow [<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( -L + 3L^\\prime \\lt 2f(x) \\lt L + L^\\prime)<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\wedge<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">( L + L^\\prime \\lt 2f(x) \\lt 3L - L^\\prime )<\/span><\/span><br \/>\n<\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ])<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\lt L^\\prime\\}\\vdash \\bot <\/span><\/span>; Ex(1,2,6,7)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\cup\\{L\\gt L^\\prime\\}\\vdash \\bot <\/span><\/span>; Eodem processu ac (8)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [(L\\lt L^\\prime) \\vee (L\\gt L^\\prime)] \\rightarrow \\bot <\/span><\/span>; <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vee<\/span><\/span>-int(8,9)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash [L\\ \\neq L^\\prime] \\rightarrow \\bot <\/span><\/span>; Def(10)\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\bot <\/span><\/span>; <strong>Modus Ponens<\/strong>(3,11)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime, L\\neq L^\\prime\\right\\} \\vdash \\bot <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\" text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash \\neg(L\\neq L^\\prime) <\/span><\/span>; <strong>Contradictio<\/strong>(12)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\" text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}f(x) = L^\\prime \\right\\} \\vdash L = L^\\prime.<\/span><\/span>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex hac demonstratione colligimus: si duo limites exsistunt, tunc sunt aequales, et ideo limes est unicus.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Algebra Limitum<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2011s\" rel=\"noopener\" target=\"_blank\"><strong>Ex iis quae hactenus tractavimus, vidimus praecipua de notione mathematica limi\u00adtis.<\/strong><\/a> Verumtamen hoc solum non sufficit ad limites computandos, nequaquam; solum insanus cruciatuum cupidus definitionem limi\u00adtis ad hunc finem adhiberet. Ad hanc difficultatem solvendam, nunc tractabimus artes quae nos adiuvabunt ad quosdam limites computandos incipiendo.<\/p>\n<p style=\"text-align: justify;\">Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_0, \\alpha, \\beta, L, M \\in \\mathbb{R},<\/span><\/span> et sint f et g functiones reales tales ut:<\/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: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} g(x) = M<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Tunc satisfiunt sequentia proprietates:<\/p>\n<h4>Limes Additionis et Differentiae Functionum<\/h4>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left(\\alpha f(x) \\pm \\beta g(x) \\right) = \\alpha L \\pm \\beta M<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>Demonstratio:<\/strong><\/p>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2053s\" rel=\"noopener\" target=\"_blank\"><strong>Consideremus collectionem praemissarum<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\mathcal{H}=\\left\\{\\lim_{x\\to x_0} f(x) = L, \\lim_{x\\to x_0} g(x) = M \\right\\}<\/span><\/span>, ex qua possumus sequentem rationem construere:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x) = L <\/span><\/span>; Praesumptio\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L|\\lt \\epsilon \\right) <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha||f(x) - L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt|x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt |\\alpha|\\epsilon \\right) <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon}:= |\\alpha|\\epsilon <\/span><\/span>; Def.\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L|\\lt \\overline{\\epsilon} \\right) <\/span><\/span>; Ex(1,2)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\alpha f(x) = \\alpha L <\/span><\/span><br \/>\n<\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}g(x) = M <\/span><\/span>; Praesumptio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}\\beta g(x) = \\beta M <\/span><\/span><br \/>\n<\/span>; Analogia ad (3)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left( 0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\beta g(x) - \\beta M|\\lt \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow \\left[|\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M|\\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right] \\right) <\/span><\/span>; Ex(3,5)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\leq |\\alpha f(x) - \\alpha L|+ |\\beta g(x) - \\beta M| <\/span><\/span>; Inequalitas Triangularis: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x,y\\in\\mathbb{R})(|x+y|\\leq |x|+|y|)<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon},\\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) - \\alpha L + \\beta g(x) - \\beta M| \\lt \\overline{\\epsilon}+ \\overline{\\overline{\\epsilon}} \\right) <\/span><\/span>; ex(6,7)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon^* := \\overline{\\epsilon} + \\overline{\\overline{\\epsilon}}<\/span><\/span>; Definitio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\epsilon^* \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |\\alpha f(x) + \\beta g(x) - \\alpha L - \\beta M| \\lt \\epsilon^* \\right) <\/span><\/span>; ex(8,9)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\beta g(x)) = \\alpha L + \\beta M <\/span><\/span><br \/>\n<\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\gamma:= - \\beta<\/span><\/span>; Definitio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) + \\gamma g(x)) = \\alpha L + \\gamma M <\/span><\/span>; Analogia(10)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span style=\"background-color: #ffff80; color:#000000\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) - \\beta g(x)) = \\alpha L - \\beta M <\/span><\/span><br \/>\n<\/span>; ex(11,12)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} (\\alpha f(x) \\pm \\beta g(x)) = \\alpha L \\pm \\beta M <\/span><\/span>; ex(10,13)\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Limes Producti Functionum<\/h4>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\left( f(x) g(x) \\right) = L M<\/span><\/span><\/p>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=2611s\" rel=\"noopener\" target=\"_blank\"><strong>Haec demonstratio paulo difficilior est quam praecedens,<\/strong><\/a> sed nihil est quod non possimus solvere per nonnullos dolos draconianos. Utentes eodem praemissarum systemate <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{H}<\/span><\/span> quo usi sumus in demonstratione praeterita, rationem sequentem construere possumus:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\epsilon} := \\frac{|\\epsilon|}{2(|M|+1)} \\leq \\frac{|\\epsilon|}{2} <\/span><\/span>; Definitio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} f(x) = L <\/span><\/span>; Praesumptio\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\epsilon} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x) - L| \\lt \\overline{\\epsilon} = \\frac{|\\epsilon|}{2(|M|+1)}\\right) <\/span><\/span>; Ex (1)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\overline{\\overline{\\epsilon}} := \\frac{|\\epsilon|}{2(|L|+1)} \\leq \\frac{|\\epsilon|}{2}<\/span><\/span>; Definitio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0} g(x) = M <\/span><\/span>; Praesumptio\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left(\\forall \\overline{\\overline{\\epsilon}} \\gt 0 \\right)\\left(\\exists \\delta \\gt 0 \\right)\\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} = \\frac{|\\epsilon|}{2(|L|+1)}\\right) <\/span><\/span>; Ex (3)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)| - |L| \\lt <\/span><\/span><br \/>\n<span style=\"background-color: #a0ffff; color:#000000;\"><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x) - L| \\lt \\overline{\\epsilon} \\lt 1 <\/span><\/span><br \/>\n<\/span>; Inequalitas Triangularis + Casus specialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\epsilon}<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)|\\lt 1 + |L| <\/span><\/span>; Ex (5)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| - |M| \\lt |g(x) - M| \\lt \\overline{\\overline{\\epsilon}} \\lt 1 <\/span><\/span>; Inequalitas Triangularis + Casus specialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\overline{\\overline{\\epsilon}}<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |g(x)| \\lt 1 + |M| <\/span><\/span>; Ex (7)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(9)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)g(x) - Mf(x)<\/span><\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ Mf(x) - LM<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; Additio zeri\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|=|<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+ M (f(x) - L)<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; Factorisatio\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(10)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq |<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)(g(x) - M)<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">| + |<\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M (f(x) - L)<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|<\/span><\/span>; Inequalitas Triangularis (9)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\leq <\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|f(x)||g(x) - M|<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/span><br \/>\n<span style=\"background-color: #a0a0ff; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M| |f(x) - L|<\/span><\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(11)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash |f(x)g(x) - LM|\\lt <\/span><\/span><br \/>\n<span style=\"background-color: #a0ffa0; color:#000000\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1 + |L|)|g(x) - M|<\/span><\/span><\/span><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">+<\/span><\/span><br \/>\n<span style=\"background-color: #a0ffff;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|M|\\overline{\\epsilon}<\/span><\/span><\/span>; Ex (5,6,10)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(12)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\overline{\\overline{\\epsilon}} + |M|\\overline{\\epsilon}\\right]<\/span><\/span>; Ex (11)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(13)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt (1+|L|)\\frac{|\\epsilon|}{2(|L|+1)} + |M|\\frac{|\\epsilon|}{2(|M|+1)}\\right]<\/span><\/span>; Ex(1,3,12)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ (1+|L|)|g(x) - M| + |M|\\overline{\\epsilon} \\lt \\frac{|\\epsilon|}{2} + \\frac{|\\epsilon||M|}{2(|M|+1)} \\lt \\frac{|\\epsilon|}{2}+ \\frac{|\\epsilon|}{2} = |\\epsilon| \\right]<\/span><\/span>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(14)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\left[ |g(x) - M|\\lt \\overline{\\overline{\\epsilon}} \\right] \\rightarrow \\left[ |f(x)g(x) - LM|\\lt |\\epsilon| \\right]<\/span><\/span>; Ex(11,13)\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(15)<\/span><\/span><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash (\\forall \\epsilon \\gt 0 ) (\\exists \\delta \\gt 0 ) \\left(0 \\lt |x-x_0|\\lt \\delta \\rightarrow |f(x)g(x) - LM|\\lt |\\epsilon| \\leq \\epsilon \\right) <\/span><\/span>; Ex(1,2,4,14)\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\">\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}\\vdash \\lim_{x\\to x_0}f(x)g(x) = LM.<\/span><\/span>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Limes Functionis Constantis<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3450s\" rel=\"noopener\" target=\"_blank\"><strong>Limes functionis constantis<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x)=c<\/span><\/span>, est ipsa constans <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span>. Id est:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>Demonstratio<\/strong><\/p>\n<p style=\"text-align: justify; \">Haec demonstratio revera simplex est, cum sit tautologia. Notum est enim quod:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c := (\\forall\\epsilon\\gt 0) (\\exists \\delta \\gt 0)(0\\lt|x-x_0|\\lt \\delta \\rightarrow |c-c|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Sed constat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0=|c-c|\\lt \\epsilon<\/span><\/span> esse tautologiam pro omni epsilon positivo, unde etiam implicatio est tautologia, ac proinde expressio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}c = c <\/span><\/span> est tautologia.<\/p>\n<h4>Limes Rationis Inter Functiones<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3563s\" rel=\"noopener\" target=\"_blank\"><strong>Nunc parati sumus ad demonstrandum regulam limi\u00adtis rationis inter duas functiones.<\/strong><\/a> Haec est:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{f(x)}{g(x)}= \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Ubi, sicut in proprietatibus prioribus, praemissarum systema supponitur:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\mathcal{H}=\\{\\lim_{x\\to x_0}f(x) = L, \\lim_{x\\to x_0}g(x) = M\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>Demonstratio<\/strong><\/p>\n<p style=\"text-align: justify; \">Fortunate, non amplius opus est demonstrationibus tam laboriosis, quia iam possumus directe uti praecedentibus resultatis. Sed antea demonstremus primum quod:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}\\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Ad hoc probandum sufficit uti regula limi\u00adtis producti una cum limite functionis constantis, modo caveamus ne <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x)<\/span><\/span> sit nulla:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1 = \\lim_{x\\to x_0}\\left( 1 \\right) \\lim_{x\\to x_0}\\left( g(x) \\cdot \\frac{1}{g(x)} \\right) = \\lim_{x\\to x_0}g(x) \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)} = M \\cdot \\lim_{x\\to x_0} \\frac{1}{g(x)}<\/span><\/span><\/p>\n<p style=\"text-align: center; \">Ergo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{1}{g(x)} = \\frac{1}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Denique, per regulam limi\u00adtis producti habemus:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\frac{f(x)}{g(x)} = \\lim_{x\\to x_0} f(x) \\frac{1}{g(x)}= L \\cdot\\frac{1}{M} = \\frac{L}{M}<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Hoc valebit dummodo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">M<\/span><\/span> non sit zero.<\/p>\n<h4>Limes Potentiae Naturalis<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3725s\" rel=\"noopener\" target=\"_blank\"><strong>Haec proprietas nos docet quod,<\/strong><\/a> si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x_0 \\to x_0}f(x) = L<\/span><\/span>, tunc sequitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right)<\/span><\/span>. Hoc per inductionem mathematicam probari potest.<\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>Demonstratio:<\/strong><\/p>\n<ul style=\"text-align: justify; \">\n<li><strong>Casus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=1<\/span><\/span>:<\/strong> (basis inductionis)\n<p style=\"text-align: justify;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^1 = \\lim_{x\\to x_0} f(x) = L.<\/span><\/span> Quod demonstrat basim inductionis \u2705<\/p>\n<\/li>\n<li><strong>Casus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n=k<\/span><\/span>:<\/strong> (passus inductivus)\n<p style=\"text-align: justify;\">Supponamus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^k = L^k <\/span><\/span> (Hypothesis Inductionis). Ostendemus tunc <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L^{k+1} <\/span><\/span>.<\/p>\n<p style=\"text-align: justify;\">Habemus: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = \\lim_{x\\to x_0} \\{f(x) [f(x)]^k\\} = \\lim_{x\\to x_0}f(x) \\lim_{x\\to x_0} [f(x)]^{k} =L \\lim_{x\\to x_0} [f(x)]^{k}<\/span><\/span>. Hoc ex regula limi\u00adtis producti, supra probata.<\/p>\n<p style=\"text-align: justify;\">Ex hypothesi inductionis sequitur <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} [f(x)]^{k+1} = L \\lim_{x\\to x_0} [f(x)]^{k} =L\\cdot L^k = L^{k+1}.<\/span><\/span> Quod complet passum inductivum \u2705<\/p>\n<\/li>\n<li>Ergo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\left( [f(x)]^n \\right) = L^n \\right). <\/span><\/span><\/li>\n<\/ul>\n<h4>Limes Radicis n-\u00e9simae<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=3912s\" rel=\"noopener\" target=\"_blank\"><strong>Similiter ac pro potentia habemus<\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall n \\in \\mathbb{N}\\right) \\left( \\lim_{x\\to x_0} \\sqrt[n]{f(x)} = \\sqrt[n]{L} \\right) <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000080;\"><strong>Demonstratio:<\/strong><\/p>\n<p style=\"text-align: justify; \">Utentes regula potentiae, quam modo probavimus, habemus:<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L= \\lim_{x\\to x_0} f(x)=\\lim_{x\\to x_0} \\left[\\sqrt[n]{f(x)}\\right]^n = \\left[ \\lim_{x\\to x_0} \\sqrt[n]{f(x)}\\right]^n <\/span><\/span><\/p>\n<p style=\"text-align: center; \">Ergo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0} \\sqrt[n]{f(x)} =\\sqrt[n]{L}.<\/span><\/span><\/p>\n<h4>Limes Potentiarum Fractionariarum<\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4007s\" rel=\"noopener\" target=\"_blank\"><strong>Cum virtutibus coniunctis duarum demonstrationum praecedentium<\/strong><\/a> concludere possumus ultimam nostram demonstrationem: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall p,q\\neq 0 \\in \\mathbb{Z}\\right) \\left( \\lim_{x\\to x_0} \\left[f(x)\\right]^{\\frac{p}{q}} = L^{\\frac{p}{q}} \\right). <\/span><\/span>, quae oritur ex regula producti quia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle [f(x)]^{\\frac{p}{q}} =[\\sqrt[q]{f(x)}]^p <\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle L^{\\frac{p}{q}} =[\\sqrt[q]{L}]^p. <\/span><\/span><\/p>\n<h4>Limes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span><\/h4>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4073s\" rel=\"noopener\" target=\"_blank\"><strong>Hac hac demonstratione clausulam huic fluctui demonstrationum imponimus,<\/strong><\/a> quibus una cum superioribus possimus posthac multos limites fere intuitive computare.<\/p>\n<p style=\"text-align: justify; \">Facile est probare <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0<\/span><\/span>, quia ut hoc valeat, necesse est ut<\/p>\n<p style=\"text-align: center; \"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall \\epsilon \\gt 0) (\\exists \\delta \\gt 0)(0\\lt |x-x_0|\\lt \\delta\\rightarrow |x-x_0|\\lt \\epsilon)<\/span><\/span><\/p>\n<p style=\"text-align: justify; \">Secundum definitionem Limites, pro omni epsilon debet existere saltem unum delta quod caeteris positis satisfaciat; unde sufficit unum tale invenire ad comprobationem. Sed hoc revera manifestum est, quoniam satis est animadvertere quodlibet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\delta\\leq\\epsilon<\/span><\/span> talem condicionem satisfacere. Ergo: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to x_0}x = x_0.<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Calculus limitum simplicium<\/h2>\n<p style=\"text-align: justify; \"><a href=\"https:\/\/www.youtube.com\/watch?v=FEPfoAfPsFY&amp;t=4155s\" rel=\"noopener\" target=\"_blank\"><strong>Gratia omnium horum theorematum, quae modo pertractavimus,<\/strong><\/a> potest ampla limitum varietas modo satis intuitivo computari, ac si simpliciter functionem evaluaremus. Ecce nonnulla exempla:<\/p>\n<ol style=\"text-align:left; \">\n<li>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{}\\\\ \\begin{array}{rl}\n\n \\displaystyle \\lim_{x\\to 2}(x^2 + 4x) &amp; = \\displaystyle \\lim_{x\\to 2}(x^2) + \\lim_{x\\to 2}(4x) \\\\ \\\\\n\n&amp; = \\displaystyle \\left(\\lim_{x\\to 2} x \\right)^2 + 4\\lim_{x\\to 2} x \\\\ \\\\\n\n&amp; = (2)^2 + 8 = 12\n\n\\end{array}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1}\\left.\\frac{(3x-1)^2}{(x+1)^3} \\right. &amp; = \\displaystyle \\frac{(3(1)-1)^2}{((1)+1)^3} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{4}{8} = \\frac{1}{2}\n\n\\end{array}\n\n<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 2} \\frac{x-2}{x^2 - 4} &amp;= \\displaystyle \\lim_{x\\to 2} \\frac{x-2}{(x-2)(x+2)} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{x\\to 2} \\frac{1}{x+2} = \\dfrac{1}{4}\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{h\\to 0} \\frac{(x+h)^3-x^3}{h} &amp;= \\displaystyle \\lim_{h\\to 0} \\frac{x^3 + 3x^2 h + 3xh^2 -x^3}{h} \\\\ \\\\\n\n&amp; = \\displaystyle\\lim_{h\\to 0} \\frac{3x^2 h + 3xh^2}{h} \\\\ \\\\\n\n&amp; = \\displaystyle \\lim_{h\\to 0} 3x^2 + 3xh = 3x^2\n\n\\end{array}\n\n <\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{} \\\\ \\begin{array}{rl}\n\n\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } &amp;=\\displaystyle \\lim_{x\\to 1} \\frac{x-1}{\\sqrt{x^2 + 3} - 2 } \\frac{\\sqrt{x^2 + 3} + 2}{\\sqrt{x^2 + 3} + 2} \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x^2 + 3) - 4 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{x^2 -1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{(x-1)(\\sqrt{x^2 + 3} + 2)}{(x-1)(x+1) } \\\\ \\\\\n\n&amp; =\\displaystyle \\lim_{x\\to 1} \\frac{\\sqrt{x^2 + 3} + 2}{ x+1 } \\\\ \\\\\n\n&amp; =\\displaystyle \\frac{2+2}{2} =2\n\n\\end{array}<\/span><\/span><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limes Functionum Variabilis Realis Summarium: In hac lectione perscrutatur definicio formalis limi\u00adnis functionum variabilis realis, ex qua demonstrantur proprietates praecipuae quae ad algebram limitum ducunt. Proposita Discendi: Post hanc lectionem discipulus poterit: Meminisse definitionem limi\u00adnis functionum variabilis realis. Demonstr\u0101re proprietates quae ad algebram limitum ducunt per deductiones . Comput\u0101re limites functionum variabilis realis utens algebra [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27786,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[1328,1298],"tags":[],"class_list":["post-33311","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>Limes Functionum Unius Variabilis Realis - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"In calculo, Limes Functionum describit rationem qua variabilis independens ad certam valor datam appropinquat\" \/>\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\/la\/limes-functionum-unius-variabilis-realis\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Limes Functionum Unius Variabilis Realis\" \/>\n<meta property=\"og:description\" content=\"In calculo, Limes Functionum describit rationem qua variabilis independens ad certam valor datam appropinquat\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/limes-functionum-unius-variabilis-realis\/\" \/>\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=\"2024-08-11T13:00:29+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-21T21:00:41+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limites.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Limes Functionum Unius Variabilis Realis\" \/>\n<meta name=\"twitter:description\" content=\"In calculo, Limes Functionum describit rationem qua variabilis independens ad certam valor datam appropinquat\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/limites.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=\"12 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Limes Functionum Unius Variabilis Realis\",\"datePublished\":\"2024-08-11T13:00:29+00:00\",\"dateModified\":\"2025-07-21T21:00:41+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/\"},\"wordCount\":3715,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2024\\\/08\\\/limites.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/limes-functionum-unius-variabilis-realis\\\/\",\"name\":\"Limes Functionum Unius Variabilis Realis - 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