{"id":34318,"date":"2024-11-18T13:00:56","date_gmt":"2024-11-18T13:00:56","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34318"},"modified":"2025-09-07T02:00:53","modified_gmt":"2025-09-07T02:00:53","slug":"asymptotae-limites-et-technica-repraesentationis-graphicae","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/","title":{"rendered":"Asymptotae, Limites et Technica Repraesentationis Graphicae"},"content":{"rendered":"<style>\np {\ntext-align:justify;\n}\n<\/style>\n<h1 style=\"text-align:center;\">Asymptotae, Limites et Technica Repraesentationis Graphicae<\/h1>\n<p style=\"text-align:center;\"><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione tractantur notiones asymptotarum et terminorum dominantium in analysi functionum. Exquiruntur asymptotae horizontales, quae describunt rationem functionis cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum; asymptotae verticales, quae significant limites infinitos cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> appropinquat ad certos valores; et asymptotae obliquae, pertinentes in functionibus rationalibus cum gradus numeratoriis superat gradum denominatorii. Item consideratur terminus dominans functionis, qui praebet approximationem pro valoribus magnis vel prope certos punctos <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi<\/strong><br \/>\nPost hanc lectionem, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Intelligere<\/strong> notionem asymptotarum horizontalium eiusque applicationem in analysi rationis functionum cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum.<\/li>\n<li><strong>Ag\u00adnoscere<\/strong> condiciones ad existentiam asymptotarum verticalium et eas applicare in studio functionum cum limitibus infinitis cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> appropinquat ad certos valores.<\/li>\n<li><strong>Analy\u00adsare<\/strong> apparentiam asymptotarum obliquarum in functionibus rationalibus cum gradus numeratoriis superat gradum denominatorii.<\/li>\n<li><strong>Applicare<\/strong> notionem termini dominantis ad approximandam rationem functionum in valoribus magnis <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> vel prope certos punctos.<\/li>\n<li><strong>Explicare<\/strong> quomodo analysis asymptotarum et terminorum dominantium conferat ad intellegendum habitum generalem functionum.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">Introductio<\/a><br \/>\n<a href=\"#2\">Asymptotae horizontales et limites ad infinitum<\/a><br \/>\n<a href=\"#3\">Asymptotae verticales et limites infiniti<\/a><br \/>\n<a href=\"#4\">Asymptotae obliquae, curvae et termini dominantes<\/a><br \/>\n<a href=\"#5\">Exercitia Soluta<\/a><br \/>\n<a href=\"#6\">Exercitia Proposita<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Ekd0oSvMbfE\" title=\"YouTube video player\" 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><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=98s\" target=\"_blank\" rel=\"noopener\"><strong>Limites quos hactenus consideravimus<\/strong><\/a> nobis concedunt definire quaedam notiones utiles ad intellegendum habitum globalem functionum, hae sunt termini dominantes et asymptotae horizontales et verticales; hae sunt, ut ita dicamus, curvae ad quas graphice functionis tendit appropinquare quam prope velimus, dum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad certum valorem.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Asymptotae horizontales et limites ad infinitum<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=137s\" target=\"_blank\" rel=\"noopener\"><strong>Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> est functio definita in<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">]a,+\\infty[<\/span>, pro aliquo <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span>, tunc est possibilitas computandi limitem <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum. Si talis limes existit, tum ex eo definitur <strong>asymptota horizontalis ad dexteram<\/strong> sicut recta aequationis<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">A_+(x) = L^+<\/span>\n<p>ubi<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to+\\infty}f(x) = L^+<\/span>\n<p>Similiter definitur <strong>asymptota horizontalis ad sinistram<\/strong> recta aequationis<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">A_-(x) = L^-<\/span>\n<p>cum<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to-\\infty}f(x) = L^-<\/span>\n<p>Asymptotae horizontales adiuvant describere habitum functionis <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cum valores <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> sine fine crescunt.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ckBGkFWse2w\/YH1GWClIciI\/AAAAAAAAE6s\/zZ_se7yShqMLiEHKNT_jkgAWuK9cme5wwCLcBGAsYHQ\/s0\/as%25C3%25ADntotahorizontal.PNG\" alt=\"asymptotae horizontales\" class=\"aligncenter lazyload\" width=\"478\" height=\"290\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ckBGkFWse2w\/YH1GWClIciI\/AAAAAAAAE6s\/zZ_se7yShqMLiEHKNT_jkgAWuK9cme5wwCLcBGAsYHQ\/s0\/as%25C3%25ADntotahorizontal.PNG\" alt=\"asymptotae horizontales\" class=\"aligncenter lazyload\" width=\"478\" height=\"290\" \/><\/noscript><br \/>\n<a name=\"3\"><\/a><\/p>\n<h2>Asymptotae verticales et limites infiniti<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=277s\" target=\"_blank\" rel=\"noopener\"><strong>Similiter ac asymptotae horizontales,<\/strong><\/a> definiuntur <strong>asymptotae verticales sursum<\/strong> functionis <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> ut recta aequationis <span class=\"katex-eq\" data-katex-display=\"false\">x=a<\/span> cum<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = +\\infty<\/span>\n<p>Et asymptota erit verticalis deorsum si<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = -\\infty<\/span>\n<p>Et secundum rationem limitum lateralium, asymptotae erunt a dextra vel a sinistra prout convenit.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ptEipMpyIhc\/YH1VDclyMxI\/AAAAAAAAE60\/LmzpK2HAU7oLpswJQy5_TLIv9jSf9whDwCLcBGAsYHQ\/s0\/asintotavertical.PNG\" alt=\"Asymptota Verticalis\" class=\" aligncenter lazyload\" width=\"428\" height=\"283\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ptEipMpyIhc\/YH1VDclyMxI\/AAAAAAAAE60\/LmzpK2HAU7oLpswJQy5_TLIv9jSf9whDwCLcBGAsYHQ\/s0\/asintotavertical.PNG\" alt=\"Asymptota Verticalis\" class=\" aligncenter lazyload\" width=\"428\" height=\"283\" \/><\/noscript><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Asymptotae obliquae, curvae et termini dominantes<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=400s\" target=\"_blank\" rel=\"noopener\"><strong>Apparitio simplicissima<\/strong><\/a> <strong>asymptotarum obliquarum<\/strong> fit cum tractamus functiones rationales<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{P(x)}{Q(x)}<\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> sunt polynomia. Cum gradus <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> maior sit quam gradus <span class=\"katex-eq\" data-katex-display=\"false\">Q(x),<\/span> possibile est facere divisionem polynomialem, quae dat eventum formae<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{P(x)}{Q(x)} = C(x) + \\dfrac{r(x)}{Q(x)}<\/span>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> est quotiens divisionis et <span class=\"katex-eq\" data-katex-display=\"false\">r(x)<\/span> est residuum. Si <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> habet gradum qui superat gradum <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> unitate, tunc fiet ut <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> sit gradus 1, id est, formam rectae habeat et dicetur esse asymptota obliqua <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span>.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-wRTmSl2Z3HE\/YH1dSl-noDI\/AAAAAAAAE68\/og2lPX_ydUUGlxYnn5hgj2mNCSeAPoQKACLcBGAsYHQ\/s0\/asintotaoblicua.PNG\" alt=\"Asymptota Obliqua\" class=\" aligncenter lazyload\" width=\"404\" height=\"239\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-wRTmSl2Z3HE\/YH1dSl-noDI\/AAAAAAAAE68\/og2lPX_ydUUGlxYnn5hgj2mNCSeAPoQKACLcBGAsYHQ\/s0\/asintotaoblicua.PNG\" alt=\"Asymptota Obliqua\" class=\" aligncenter lazyload\" width=\"404\" height=\"239\" \/><\/noscript><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=Ekd0oSvMbfE&amp;t=633s\" target=\"_blank\" rel=\"noopener\"><strong>Si generaliter, <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> habet gradum<\/strong><\/a> qui superat gradum <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span> quacumque magnitudine, tunc fiet ut <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> habeat gradum aequalem differentiae graduum inter <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)<\/span>, atque erit consequenter curva polynomialis in genere. Hoc in casu non solet dici <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> esse asymptotam, quamvis habitus generalis <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> sit \u201casymptotice appropinquare\u201d ad <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> dum <span class=\"katex-eq\" data-katex-display=\"false\">x\\to\\pm\\infty<\/span>. Hoc in casu dicitur <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> esse <strong>terminus dominans <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> pro magnis valoribus <span class=\"katex-eq\" data-katex-display=\"false\">x.<\/span><\/strong><\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-IEYO071tTuY\/YH1fjR-WWnI\/AAAAAAAAE7E\/ga2rZ02i8QU5R1IMvQB9rpgFuDknAGfbACLcBGAsYHQ\/s0\/terminoDominante.PNG\" alt=\"Terminus Dominans et Asymptota Verticalis\" class=\" aligncenter lazyload\" width=\"479\" height=\"437\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-IEYO071tTuY\/YH1fjR-WWnI\/AAAAAAAAE7E\/ga2rZ02i8QU5R1IMvQB9rpgFuDknAGfbACLcBGAsYHQ\/s0\/terminoDominante.PNG\" alt=\"Terminus Dominans et Asymptota Verticalis\" class=\" aligncenter lazyload\" width=\"479\" height=\"437\" \/><\/noscript><\/p>\n<p>Item possibile est loqui de termino dominante cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est prope <span class=\"katex-eq\" data-katex-display=\"false\">a\\in\\mathbb{R}<\/span>.<\/p>\n<p>Si <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = P(x)\/Q(x) = C(x) + r(x)\/Q(x),<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">P(x),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x),<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">r(x)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C(x)<\/span> sint polynomia. Si <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to a}f(x) = \\infty,<\/span> tum dicetur quotiens <span class=\"katex-eq\" data-katex-display=\"false\">r(x)\/Q(x)<\/span> esse <strong>terminus dominans <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> prope <span class=\"katex-eq\" data-katex-display=\"false\">x=a.<\/span><\/strong><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercitia Soluta<\/h2>\n<h3><strong>Exercitium 1:<\/strong><\/h3>\n<p>Determina asymptotas horizontales et verticales functionis<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\dfrac{3x + 1}{x - 2}<\/span>\n<p><strong>Solutio:<\/strong><\/p>\n<p>Ad inveniendam <strong>asymptotam horizontalem<\/strong>, computamus limitem <span class=\"katex-eq\" data-katex-display=\"false\">f(x)<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span>:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to \\pm\\infty} \\dfrac{3x + 1}{x - 2} = 3<\/span>\n<p>Ergo asymptota horizontalis est <span class=\"katex-eq\" data-katex-display=\"false\">y = 3<\/span>.<\/p>\n<p>Ad <strong>asymptotam verticalem<\/strong>, agnoscimus valorem ubi denominator evanescit, id est, cum <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span>.<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to 2^\\pm} \\dfrac{3x + 1}{x - 2} = \\pm\\infty<\/span>\n<p>Hoc significat asymptotam verticalem in <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span>.<\/p>\n<p><strong>Eventus finalis:<\/strong> Functio habet asymptotam horizontalem in <span class=\"katex-eq\" data-katex-display=\"false\">y = 3<\/span> et asymptotam verticalem in <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span>.<\/p>\n<h3><strong>Exercitium 2:<\/strong><\/h3>\n<p>Inveni asymptotas horizontales et obliquas, si existunt, functionis <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\frac{2x^2 + 3x + 4}{x + 1}<\/span>.<\/p>\n<p><strong>Solutio:<\/strong><\/p>\n<p>Primum quaerimus <strong>asymptotam horizontalem<\/strong> computando limitem cum <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span>. Cum gradus numeratorii maior sit quam gradus denominatorii, asymptota horizontalis non existit.<\/p>\n<p>Ad <strong>asymptotam obliquam<\/strong>, facimus divisionem polynomialem obtinentes hunc eventum:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\dfrac{2x^2 + 3x + 4}{x + 1} = 2x + 1 + \\dfrac{3}{x + 1}<\/span>\n<p>Itaque asymptota obliqua est recta <span class=\"katex-eq\" data-katex-display=\"false\">y = 2x + 1<\/span>, quae est terminus dominans functionis.<\/p>\n<p><strong>Eventus finalis:<\/strong> Functio non habet asymptotam horizontalem, sed habet asymptotam obliquam quae est recta <span class=\"katex-eq\" data-katex-display=\"false\">y = 2x + 1<\/span>.<\/p>\n<h3><strong>Exercitium 3:<\/strong><\/h3>\n<p>Computa asymptotam verticalem <span class=\"katex-eq\" data-katex-display=\"false\">h(x) = \\frac{5}{x^2 - 4}<\/span>.<\/p>\n<p><strong>Solutio:<\/strong><\/p>\n<p>Ad inveniendam <strong>asymptotam verticalem<\/strong>, agnoscimus valores ubi denominator evanescit, id est, <span class=\"katex-eq\" data-katex-display=\"false\">x^2 - 4 = 0<\/span>. Hoc fit in <span class=\"katex-eq\" data-katex-display=\"false\">x = \\pm 2<\/span>.<\/p>\n<p>Aestimamus limites laterales pro unoquoque valore:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to 2^\\pm} \\dfrac{5}{x^2 - 4} = \\pm\\infty<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{x \\to -2^\\pm} \\dfrac{5}{x^2 - 4} = \\pm\\infty<\/span>\n<p><strong>Eventus finalis:<\/strong> Functio habet asymptotas verticales in <span class=\"katex-eq\" data-katex-display=\"false\">x = 2<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">x = -2<\/span>.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Exercitia Proposita<\/h2>\n<ol>\n<li>Analysa functionem <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\frac{2x^2 - 3x + 1}{x^2 + x - 2}<\/span>. Determina eius asymptotas horizontales, verticales et obliquas, si existant. Explica unumquemque passum ad confirmandum conceptum asymptotarum et computum limitum.<\/li>\n<li>Evalu\u00ada functionem <span class=\"katex-eq\" data-katex-display=\"false\">g(x) = \\frac{3x^3 + 2x}{x^2 + 1}<\/span>. Identifica terminum dominantem cum <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> tendit ad infinitum. Deinde, verifica si asymptota obliqua existat, rationem reddens responsi tui.<\/li>\n<li>Designa graphice approximatam functionis <span class=\"katex-eq\" data-katex-display=\"false\">h(x) = \\frac{5x - 4}{x + 1}<\/span>. Include asymptotas horizontales, verticales et obliquas (si existant) et analysa habitum <span class=\"katex-eq\" data-katex-display=\"false\">h(x)<\/span> pro valoribus extremis <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>.<\/li>\n<li>Proba si functio <span class=\"katex-eq\" data-katex-display=\"false\">k(x) = \\frac{x^2 - 4x + 3}{x^2 - 1}<\/span> habeat asymptotas verticales. Disserere de munere terminorum dominantium in analysi limitis <span class=\"katex-eq\" data-katex-display=\"false\">k(x)<\/span> in valoribus ubi functio tendit ad infinitum.<\/li>\n<li>Explora terminos dominantes <span class=\"katex-eq\" data-katex-display=\"false\">m(x) = \\frac{2x^4 + 3x^2 - x + 5}{x^3 - x^2 + 2}<\/span>. Determina habitum <span class=\"katex-eq\" data-katex-display=\"false\">m(x)<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">x \\to \\pm\\infty<\/span>, et conclude si appropinquet ad curvam polynomialem potius quam ad rectam.<\/li>\n<li>Formula functionem rationalem ad tuam electionem et describe accurate quomodo computentur asymptotae horizontales, verticales et obliquae, praeterea termini dominantes. Exhibe inventa tua per graphicos ad visualizandum unumquemque genus asymptotae.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Asymptotae, Limites et Technica Repraesentationis Graphicae Summarium: In hac lectione tractantur notiones asymptotarum et terminorum dominantium in analysi functionum. Exquiruntur asymptotae horizontales, quae describunt rationem functionis cum tendit ad infinitum; asymptotae verticales, quae significant limites infinitos cum appropinquat ad certos valores; et asymptotae obliquae, pertinentes in functionibus rationalibus cum gradus numeratoriis superat gradum denominatorii. Item [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29511,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":86,"footnotes":""},"categories":[1328,1298],"tags":[],"class_list":["post-34318","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Asymptotae, Limites et Technica Repraesentationis Graphicae - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Haec dux te adiuvabit ad intellegendum asymptotas horizontales, verticales et obliquas, atque quomodo eas obtinere ex studio limitum.\" \/>\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\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Asymptotae, Limites et Technica Repraesentationis Graphicae\" \/>\n<meta property=\"og:description\" content=\"Haec dux te adiuvabit ad intellegendum asymptotas horizontales, verticales et obliquas, atque quomodo eas obtinere ex studio limitum.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\" \/>\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=\"2024-11-18T13:00:56+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-07T02:00:53+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/asintotas-1024x550.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Asymptotae, Limites et Technica Repraesentationis Graphicae\" \/>\n<meta name=\"twitter:description\" content=\"Haec dux te adiuvabit ad intellegendum asymptotas horizontales, verticales et obliquas, atque quomodo eas obtinere ex studio limitum.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/asintotas.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\/asymptotae-limites-et-technica-repraesentationis-graphicae\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Asymptotae, Limites et Technica Repraesentationis Graphicae\",\"datePublished\":\"2024-11-18T13:00:56+00:00\",\"dateModified\":\"2025-09-07T02:00:53+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\"},\"wordCount\":1226,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/11\/asintotas.jpg\",\"articleSection\":[\"Calculus Differentialis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\",\"url\":\"https:\/\/toposuranos.com\/material\/la\/asymptotae-limites-et-technica-repraesentationis-graphicae\/\",\"name\":\"Asymptotae, Limites et Technica Repraesentationis Graphicae - 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