{"id":66,"date":"2023-09-17T11:07:37","date_gmt":"2023-09-17T11:07:37","guid":{"rendered":"https:\/\/mathority.org\/nl\/schuine-asymptoot\/"},"modified":"2023-09-17T11:07:37","modified_gmt":"2023-09-17T11:07:37","slug":"schuine-asymptoot","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/schuine-asymptoot\/","title":{"rendered":"Schuine asymptoot"},"content":{"rendered":"<p>In dit artikel leggen we uit wat de schuine asymptoten van een functie zijn. Je leert wanneer een functie een schuine asymptoot heeft en hoe deze wordt berekend. En bovendien kun je voorbeelden van schuine asymptoten zien en oefenen met oefeningen die stap voor stap worden opgelost. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-una-asintota-oblicua\"><\/span> Wat is een schuine asymptoot?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>De schuine asymptoot van een functie is een hellende lijn die de grafiek voor onbepaalde tijd nadert zonder deze ooit te overschrijden.<\/strong> Bijgevolg zijn alle schuine asymptoten lijnen met de vergelijking <em>y=mx+n<\/em> .<\/p>\n<p> De helling en oorsprong van een schuine asymptoot worden berekend met behulp van de volgende formules: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/asymptote-oblique-dune-fonction.webp\" alt=\"schuine asymptoot van een functie\" class=\"wp-image-1362\" width=\"290\" height=\"328\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-calcular-la-asintota-oblicua-de-una-funcion\"><\/span> Hoe de schuine asymptoot van een functie te berekenen<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Om de schuine asymptoot van een functie te berekenen, moeten de volgende stappen worden uitgevoerd:<\/p>\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">Bereken de limiet tot oneindig van de functie gedeeld door x.<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Als de bovenstaande limiet resulteert in een re\u00ebel getal dat niet nul is, betekent dit dat de functie een schuine asymptoot heeft. En bovendien zal de helling van de schuine asymptoot de waarde zijn die bij de limiet wordt verkregen.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-004f6e72e10d1ba23da76d2fd8ea13f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty}\\frac{f(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">In dit geval hoeft u alleen nog maar het snijpunt van de schuine asymptoot te berekenen door de volgende limiet op te lossen:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cc74ce0447b0a9148cae947674ad085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} [f(x)-mx]\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"170\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/ol>\n<p> <strong>Let op:<\/strong> de limieten moeten worden berekend op plus en min oneindig, maar normaal gesproken geven ze hetzelfde resultaat en daarom vereenvoudigen we door \u00b1\u221e te plaatsen. Maar als de grenzen bij plus en min oneindig verschillend waren, zouden de linker schuine asymptoot en de rechter schuine asymptoot afzonderlijk moeten worden berekend. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-asintota-oblicua\"><\/span> Schuin asymptootvoorbeeld<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Vervolgens nemen we de schuine asymptoot van de volgende rationale functie, zodat u een voorbeeld kunt zien van hoe dit wordt gedaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b02f6283fd481e890a943badfa2c876f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=\\cfrac{x^2+1}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"109\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> De schuine asymptoten zijn van het type<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad313410fc976bc53709807aa8aed8e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n.\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -4px;\"><\/p>\n<p> We berekenen dus eerst de helling van de lijn<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p> met de bijbehorende formule:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b50ee5cbc3cf33f7fd42c3fe03a3d71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-870fc158a1aabb54cb5f3b4296381512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m= \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+1}{x}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"141\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Om deze limiet op te lossen moeten we de eigenschappen van breuken toepassen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f313d826cd1a2dd1ef66b1d0a40efb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{\\cfrac{a}{b}}{\\cfrac{c}{d}}=\\cfrac{a\\cdot d}{b\\cdot c}\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"69\" style=\"vertical-align: -39px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dfc6b6aa917846535c6c4b6158961988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m= \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+1}{x}}{x}=\\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"264\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> En nu berekenen we de limiet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-542fc353481ddc465b7a40f665d3661d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1}{x^2} = \\cfrac{+\\infty}{+\\infty} = \\cfrac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"269\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> In dit geval is het resultaat van de onbepaaldheid van oneindigheid tussen oneindigheid de verdeling van de co\u00ebffici\u00ebnten van x van de hoogste graad, aangezien de teller en de noemer van dezelfde orde zijn.<\/p>\n<p> De bovenstaande limiet geeft een re\u00ebel getal dat niet nul is, dus de functie heeft een schuine asymptoot. We zullen nu het y-snijpunt berekenen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> van de asymptoot met behulp van de overeenkomstige formule:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-45119c7a74d77a92d7a6cfd5b5c3544f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"173\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9197669cc0e41aa22224b552b21b31ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"191\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> We proberen de limiet te berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4d7fa012eace37e82c243012c91f1a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-x\\right] = \\cfrac{+\\infty}{+\\infty} - (+\\infty) = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"412\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Maar we krijgen onbepaaldheid oneindig min oneindig. Het is daarom noodzakelijk om de termen terug te brengen tot een gemeenschappelijke noemer. Om dit te doen, vermenigvuldigen en delen we de x door de noemer van de breuk:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2355ed9411470b9fd20a50ebbd48726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n=\\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x\\cdot x}{x} \\right] = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x^2}{x}\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"391\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> Nu de twee termen dezelfde noemer hebben, kunnen we ze groeperen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f932ebc8728669c7c6b57e115c444fc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\cfrac{x^2+1}{x}-\\cfrac{x^2}{x} \\right] =  \\lim_{x \\to \\pm\\infty} \\cfrac{x^2+1-x^2}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"358\" style=\"vertical-align: -23px;\"><\/p>\n<\/p>\n<p> We werken met de teller:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c39259f829c9e99fc88819c6ae266e82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty}  \\cfrac{\\phantom{2}1\\phantom{2}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"112\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> En ten slotte lossen we de limiet op:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16a0044416d02e77b05f65f1bb93d4cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty}  \\cfrac{\\phantom{2}1\\phantom{2}}{x}= \\cfrac{1}{\\pm\\infty} = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"201\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Dus <em>n<\/em> =0. Daarom is de schuine asymptoot een lineaire functie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a68ac5c51acd0f68bd022aee64cd9cd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x+0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4909df7491ef54f0df1e922bc29417f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=x}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> De bestudeerde functie wordt weergegeven in de onderstaande grafiek. Zoals je kunt zien, komt de functie heel dicht bij de lijn y=x, maar raakt deze nooit, omdat deze een schuine asymptoot is: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/image-1.png\" alt=\"voorbeeld van schuine asymptoot\" class=\"wp-image-1374\" width=\"424\" height=\"478\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-asintotas-oblicuas\"><\/span> Opgeloste oefeningen op schuine asymptoten<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Zoek de schuine asymptoot van de volgende rationale functie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ecc70adc78bf259cf6e36c0dcf1bee7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)= \\frac{x^2+2x+3}{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"150\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De schuine asymptoten hebben de vorm<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e4adcc4368f6296906b6231bf17a6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<p> is het daarom noodzakelijk om de parameters <em>m<\/em> en <em>n<\/em> te berekenen. We berekenen eerst <em>m<\/em> door de formule toe te passen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc38f695cee95c4c60c6e2591345119e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x} = \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{x^2+2x+3}{x+1}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"293\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vereenvoudigen de breuk door de eigenschappen van breuken toe te passen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef59ac0cd51c39c615896543993c12b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{(x+1)\\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"180\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c51e373fd07a821f8e75d63e38f252dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{x^2+x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"180\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we lossen de limiet op:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-653fa714bca94b5cc4f3ed715d7c1520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3}{x^2+x}= \\frac{+\\infty}{+\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"308\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus <em>m<\/em> =1. Laten we nu het snijpunt van de schuine asymptoot berekenen door de formule ervan toe te passen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e779b5ac239ae56c53427510dbd54dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-1x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"395\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We proberen de limiet te berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f95f290fbf258d45aa5765008d7aad13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-x\\right]= \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"320\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we krijgen de onbepaalde vorm oneindig min oneindig. We moeten daarom de termen terugbrengen tot een gemeenschappelijke noemer en ze vervolgens groeperen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0712d34ed442d9e12ef2490f04df078a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-x\\right] =\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-\\frac{x \\cdot (x+1)}{x+1} \\right] = \\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{x^2+2x+3}{x+1}-\\frac{x^2+x}{x+1} \\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3-(x^2+x)}{x+1}\\\\[6ex]\\displaystyle =\\lim_{x \\to \\pm\\infty} \\frac{x^2+2x+3-x^2-x}{x+1}=\\\\[6ex]=\\displaystyle \\lim_{x \\to \\pm\\infty} \\frac{x+3}{x+1}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"434\" width=\"300\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte lossen we de limiet op:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee7e1fdd8e781abed322fed1182ddb15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to \\pm\\infty} \\frac{x+3}{x+1} = \\frac{\\infty}{\\infty} = \\frac{1}{1} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"241\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kortom, de schuine asymptoot van de functie is: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-69c0f50795c1f6034c0cd04201f614d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 1x + 1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ffe94db5ae8fa1abc72e6007c2c0586_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y = x + 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 2<\/h3>\n<p> Vind alle schuine asymptoten van de volgende rationale functie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-144807b8c72afbd43bb3f97d69cedb35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{2x^2-5}{x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"118\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Zie de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst gebruiken we de formule voor de helling van de schuine asymptoot:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc900ded359235b2293ec151e715daea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m = \\lim_{x \\to \\pm\\infty} \\frac{f(x)}{x} = \\lim_{x \\to \\pm\\infty} \\cfrac{\\cfrac{2x^2-5}{x+3}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"261\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vereenvoudigen de breuk door de eigenschappen van breuken toe te passen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c5afa9b1ca5f1c73e6b8e64c8fb9420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{(x+3)\\cdot x}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"168\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03a4b53a445bded103e8de4404620693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{x^2+3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"149\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we bepalen de limiet:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-461c274fc210474eddaf061463e92aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5}{x^2+3x}= \\frac{+\\infty}{+\\infty} = \\frac{2}{1} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"278\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De limiet geeft een ander re\u00ebel getal dan nul, dus het is een rationale functie met een schuine asymptoot waarvan de helling 2 is.<\/p>\n<p class=\"has-text-align-left\"> Laten we nu het snijpunt berekenen door de overeenkomstige formule toe te passen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a04a1abaebfc5e1781dd7d98399888e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[f(x)-mx\\right] = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"364\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We proberen de limiet te berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00f35703d153fe6911328d143588e1cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]= \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"298\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we verkrijgen het verschil-onbepaaldheid van oneindigheden. Daarom reduceren we de termen tot een gemeenschappelijke noemer en gaan we als volgt te werk:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4920e8b21b180c4f2740ce712d9f30d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle n = \\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-2x\\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[\\frac{2x^2-5}{x+3}-\\frac{2x\\cdot (x+3)}{x+3} \\right] = \\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty} \\left[ \\frac{2x^2-5}{x+3}-\\frac{2x^2+6x}{x+3}\\right]=\\\\[6ex]=\\displaystyle\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5-(2x^2+6x)}{x+3}\\\\[6ex]\\displaystyle =\\lim_{x \\to \\pm\\infty}\\frac{2x^2-5-2x^2-6x}{x+3}=\\\\[6ex]=\\displaystyle \\lim_{x \\to \\pm\\infty} \\frac{-6x-5}{x+3}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"434\" width=\"277\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte lossen we de limiet op:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00b75da44399a44a4e215fd4baccf214_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle n =\\lim_{x \\to \\pm\\infty} \\frac{-6x-5}{x+3}= \\frac{\\infty}{\\infty}=\\frac{-6}{1} = \\bm{-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"292\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Samenvattend is de schuine asymptoot van de fractionele functie: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fbe1cc5f3362ddbd80ed0b29c0bb4ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = mx+n\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac6ac25ec7b85209d4d7d855e3d0b501_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=2x-6}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In dit artikel leggen we uit wat de schuine asymptoten van een functie zijn. Je leert wanneer een functie een schuine asymptoot heeft en hoe deze wordt berekend. En bovendien kun je voorbeelden van schuine asymptoten zien en oefenen met oefeningen die stap voor stap worden opgelost. Wat is een schuine asymptoot? De schuine asymptoot &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/schuine-asymptoot\/\"> <span class=\"screen-reader-text\">Schuine asymptoot<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[43],"tags":[],"class_list":["post-66","post","type-post","status-publish","format-standard","hentry","category-functielimieten"],"yoast_head":"<!-- This site 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