{"id":34294,"date":"2024-10-22T13:00:47","date_gmt":"2024-10-22T13:00:47","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34294"},"modified":"2025-09-07T01:40:13","modified_gmt":"2025-09-07T01:40:13","slug":"limes-ad-infinitum-definitiones-et-exempla","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/limes-ad-infinitum-definitiones-et-exempla\/","title":{"rendered":"Limes ad Infinitum: Definitiones et Exempla"},"content":{"rendered":"<p><center><\/p>\n<h1>Limes ad Infinitum: Definitiones et Exempla<\/h1>\n<p><em><\/p>\n<p><strong>Summarium:<\/strong><br \/>\nIn hac lectione tractabuntur limites ad infinitum, describens rationem <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum. Explicantur limites fundamentales ut <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} \\frac{1}{x} = 0<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x\\to \\infty} k = k<\/span>, una cum proprietatibus algebraicis similibus iis limitum finitorum.\n<\/p>\n<p><\/em><\/p>\n<p><strong>Proposita Discendi:<\/strong><br \/>\nAd finem huius lectionis, discipulus poterit<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Describere<\/strong> rationem <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum.<\/li>\n<li><strong>Definire<\/strong> limitem ad infinitum utens notatione mathematica formali.<\/li>\n<li><strong>Applicare<\/strong> proprietates algebraicas in calculo limitum ad infinitum.<\/li>\n<li><strong>Distingere<\/strong> inter varios casus limitum in functionibus rationalibus ad infinitum.<\/li>\n<li><strong>Demonstrate<\/strong> validitatem proprietatum additionis, subtractionis, multiplicationis, divisionis et potestatum limitum ad infinitum.<\/li>\n<li><strong>Resovere<\/strong> exercitationes practicas limitum ad infinitum in variis functionibus.<\/li>\n<\/ol>\n<p><u>INDEX CONTENTORUM<\/u>:<br \/>\n<a href=\"#1\">Introductio<\/a><br \/>\n<a href=\"#2\">Definitio Limitis ad Infinitum<\/a><br \/>\n<a href=\"#3\">Limites Fundamentales ad Infinitum<\/a><br \/>\n<a href=\"#4\">Algebra Limitum ad Infinitum<\/a><br \/>\n<a href=\"#5\">Limes ad infinitum in Functionibus Rationalibus<\/a><br \/>\n<a href=\"#6\">Exempla limitum ad infinitum<\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MjjSAQLeNBE\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=41s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Unus ex elementis maxime characteristicis calculi sunt infinitum et limes ad infinitum.<\/span><\/strong><\/a> Notio infiniti non spectat ad numerum realem, sed conatur describere magnitudinem quae quamlibet metam realem superat. Exempli gratia, cum habeamus functionem <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = 1\/x<\/span> et quaeramus de eius ratione cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tam magnum sit quam velimus, cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum (<span class=\"katex-eq\" data-katex-display=\"false\">x\\to \\infty<\/span>), id quod observamus est <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> posse proinde accedere ad nullum quantum velimus. Ad hoc scribimus:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to + \\infty}\\dfrac{1}{x} = 0<\/span>\n<p style=\"text-align: justify;\">Graphice, hoc negotium talem speciem habet:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite ad infinitum\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-fX0lg2ICTTU\/YGD0hNJWI6I\/AAAAAAAAEwQ\/0v8hW6ARkQYDIzT_eG5WVgZ0-pPwwPBwgCLcBGAsYHQ\/s0\/limiteAlinfinito.PNG\" alt=\"limite ad infinitum\" class=\"alignnone size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><\/center><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Definitio Limitis ad Infinitum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=144s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ex hac idea quam modo introduximus<\/span><\/strong><\/a> possumus definitionem mathematicam limitis ad infinitum formulare:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})(M\\lt x \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}f(x) = L := (\\forall\\epsilon\\gt 0) (\\exists N\\in\\mathbb{R})(x\\lt N \\rightarrow |f(x) - L|\\lt \\epsilon )<\/span>\n<p style=\"text-align: justify;\">Notio intuitiva huius limitis nobis indicat quid fiat cum <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> dum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> ab origine tam longe quam volumus recedit, sive ad dextram sive ad sinistram. Ratio ad calculum limitum ad infinitum non multum differt ab ea quam adhibemus ad limites finitos computandos, quia eius algebra fere eadem est; tantum attendere debemus ad sequentia resultata<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Limites Fundamentales ad Infinitum<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=450s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ex his definitionibus demonstrare possumus<\/span><\/strong><\/a> sequentes limites fundamentales.<\/p>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}k = k <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}\\dfrac{1}{x} = 0 <\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify;\"><span style=\"color: #000080;\">DEMONSTRATIO:<\/span><\/p>\n<ol style=\"text-align: justify;\">\n<li>Ex definitione limitis ad infinitum, habemus <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> aequivalere ad dicendum:<span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|k-k\\right|\\lt \\epsilon \\right)<\/span>Sed <span class=\"katex-eq\" data-katex-display=\"false\">\\left|k-k\\right|=0\\lt \\epsilon <\/span> semper valet et pro quolibet <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\gt 0,<\/span> nullo referto de valore <span class=\"katex-eq\" data-katex-display=\"false\">M,<\/span> unde limes confirmatur.\n<p>&nbsp;<\/li>\n<li>Scimus quod, definitione, <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}k = k <\/span> aequivalet ad dicendum:<span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\epsilon\\gt 0) (\\exists M\\in\\mathbb{R})\\left(M\\lt x \\rightarrow \\left|\\dfrac{1}{x}\\right|\\lt \\epsilon \\right)<\/span>Haec autem consequentia statim impletur si consideremus <span class=\"katex-eq\" data-katex-display=\"false\">M=1\/\\epsilon,<\/span> ita ut limes confirmetur.\n<p>&nbsp;<\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Hae demonstrationes similiter peraguntur cum <span class=\"katex-eq\" data-katex-display=\"false\">x\\to+\\infty.<\/span>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Algebra Limitum ad Infinitum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=620s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Algebra limitum infinitum similis est algebrae limitum finitorum.<\/span> <\/strong><\/a>Si <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}f(x) = L<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm \\infty}g(x) = M,<\/span> tunc sequentes regulae valent:<\/p>\n<ol style=\"text-align: justify;\">\n<li><strong>Additio et Subtractio Limitum:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}(f(x)\\pm g(x)) = L \\pm M<\/span><\/li>\n<li><strong>Multiplicatio per constantem:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}cf(x) = cL<\/span><\/li>\n<li><strong>Productum limitum:<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)g(x) = LM<\/span><\/li>\n<li><strong>Divisio Limitum:<\/strong> Dummodo <span class=\"katex-eq\" data-katex-display=\"false\">M\\neq 0,<\/span> tunc <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}f(x)\/g(x)=L\/M<\/span><\/li>\n<li><strong>Potestates Limitum:<\/strong> Si <span class=\"katex-eq\" data-katex-display=\"false\">p,q \\in\\mathbb{Z}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">q\\neq 0<\/span>, tunc <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\pm\\infty}[f(x)]^{p\/q} = L^{p\/q}<\/span>\nSi <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> est par, ponitur <span class=\"katex-eq\" data-katex-display=\"false\">L\\geq 0<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify;\">Revera, demonstratio omnium harum proprietatum est similis illis <a href=\"https:\/\/toposuranos.com\/la-definicion-de-limite-demostraciones-y-teoremas\/\" target=\"_blank\" rel=\"noopener\">limitum finitorum<\/a><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Limes ad infinitum in Functionibus Rationalibus<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=792s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Functio rationalis est illa quae exprimi potest ut quotiens inter duos polynomios.<\/span><\/strong><\/a> Cum calculum limitum ad infinitum in hoc genere functionum facimus, proprietatem animadvertere possumus quae utilissima est:<\/p>\n<p style=\"text-align: justify;\">Supponamus nos velle computare <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to \\infty}P(x)\/Q(x)<\/span>\n<ul style=\"text-align: justify;\">\n<li>Si gradus <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> maior est quam gradus <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, tunc magnitudo functionis <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> sine limite crescit cum <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\infty<\/span> (limes non exsistet).<\/li>\n<li>Cum gradus <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> minor est quam gradus <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, tunc limes erit nullus.<\/li>\n<li>Denique, si gradus <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> aequalis est gradu <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, tunc limes erit aequalis quotiens coefficientium qui comitantur potestatem gradus maioris.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Optimum huius resultati est quod, sicut in sequentibus exemplis videbimus, analogice operatur etiam si potestates implicatae non sint numeri integri.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exempla limitum ad infinitum<\/h2>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x+1}{x^2+3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=907s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^3 + 7}{x^3 - x^2 + x + 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=986s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{9x^4 + x}{2x^4 + 5x^2 - x + 6}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{10x^5 + x4 + 31}{x^4 - 7x^3 + 7x^2 + 9}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1111s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{2\\sqrt{x}+x^{-1}}{3x - 7}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1220s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to -\\infty}\\dfrac{2x^{5\/3} - x^{1\/3} + 7}{x^{8\/5}+3x + \\sqrt{x}}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1284s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{\\sqrt[3]{x}-5x+3}{2x + x^{2\/3} - 4}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1406s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty}\\dfrac{x^{8\/3}+2x + \\sqrt{x}}{x^2+x-3}<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=MjjSAQLeNBE&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Limes ad Infinitum: Definitiones et Exempla Summarium: In hac lectione tractabuntur limites ad infinitum, describens rationem cum tendit ad infinitum. Explicantur limites fundamentales ut et , una cum proprietatibus algebraicis similibus iis limitum finitorum. Proposita Discendi: Ad finem huius lectionis, discipulus poterit Describere rationem cum tendit ad infinitum. Definire limitem ad infinitum utens notatione mathematica [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29336,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[1328,1298],"tags":[],"citadela-post-location":[],"class_list":["post-34294","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 v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Limes ad Infinitum: Definitiones et Exempla - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Nesciens limites infinitos tibi puncta aufert! 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