{"id":419,"date":"2023-07-04T04:08:40","date_gmt":"2023-07-04T04:08:40","guid":{"rendered":"https:\/\/mathority.org\/nl\/onbepaaldheid-oneindig-min-oneindig-%e2%88%9e-%e2%88%9e\/"},"modified":"2023-07-04T04:08:40","modified_gmt":"2023-07-04T04:08:40","slug":"onbepaaldheid-oneindig-min-oneindig-%e2%88%9e-%e2%88%9e","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/onbepaaldheid-oneindig-min-oneindig-%e2%88%9e-%e2%88%9e\/","title":{"rendered":"Onbepaaldheid oneindig min oneindig (\u221e-\u221e)"},"content":{"rendered":"<p>In dit artikel leggen we uit hoe je de oneindige minus oneindige (\u221e-\u221e) onbepaaldheid kunt oplossen. Je vindt voorbeelden van deze onbepaaldheid met verschillende soorten functies en daarnaast kun je oefenen met oefeningen die stap voor stap worden opgelost van onbepaaldheid oneindig min oneindig. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"resolver-la-indeterminacion-infinito-menos-infinito\"><\/span> Het oplossen van onbepaaldheid oneindig min oneindig<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Wanneer de limiet van een functie oneindig min oneindig geeft, betekent dit dat het een onbepaaldheid is (of een onbepaalde vorm). Dat wil zeggen <strong>dat de limiet van een functie die onbepaaldheid minus oneindig geeft,<\/strong> niet kan worden bepaald door de directe berekening uit te voeren, maar dat er eerder een voorbereidende procedure moet worden uitgevoerd.<\/p>\n<p> Om <strong>de oneindige minus oneindige onbepaaldheid op te lossen,<\/strong> moeten we daarom eerst een procedure toepassen die afhangt van het type functie: als het een polynomiale functie is, kan deze door vergelijking worden berekend, als het een rationale functie is, moeten breuken worden gereduceerd tot een gemeenschappelijke noemer, en als het een irrationele functie is, moet deze worden vermenigvuldigd met de conjugaat.<\/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-b1bfc56d2079c86e8ad6e1943311b730_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\Bigl(f(x)-g(x)\\Bigr)=\\infty-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"236\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Vervolgens zullen we met voorbeelden zien hoe de onbepaaldheid oneindig min oneindig in elk type functie wordt opgelost. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-en-funciones-polinomicas\"><\/span> Oneindige minus oneindige onbepaaldheid in polynomiale functies <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> In een polynoom is de onbepaaldheid oneindig minus oneindig gelijk aan de hoogste orde oneindigheid, dat wil zeggen dat de term van de hoogste orde het positieve of negatieve teken van de oneindigheid bepaalt.<\/p>\n<\/div>\n<p> Kijk bijvoorbeeld naar de limiet van de volgende polynoomfunctie die de onbepaalde vorm oneindig minus oneindig geeft:<\/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-f61115731d29dc9f05941968417c9443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^2-3x\\bigr)=(+\\infty)^2-3\\cdot (\\infty)=+\\infty-\\infty=+\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"421\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> In dit geval is de term x <sup>2<\/sup> van de tweede graad en de term 3x van de eerste graad, dus de monomial x <sup>2<\/sup> is dominant omdat deze van hogere orde is. Daarom is het resultaat van de limiet oneindig, verkregen uit deze term.<\/p>\n<p> Kijk eens naar deze andere voorbeelden:<\/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-3bfb5cd294a19de382f74738af6be724_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^5-4x^2-3x\\bigr)=(+\\infty)^5=+\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to-\\infty}\\bigl(-3x^2-5x\\bigr)=-3\\cdot (-\\infty)^2=-3\\cdot \\infty=-\\infty\\\\[5ex]\\displaystyle\\lim_{x\\to+\\infty}\\bigl(x^7-5x^4+x^3-2x-10\\bigr)=(+\\infty)^7=+\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"387\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kortom, als we grenzen stellen aan oneindigheid in polynomiale functies <strong>, moeten we oneindigheid simpelweg vervangen door de term van de hoogste graad<\/strong> , waarbij we alle andere termen negeren. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-fracciones\"><\/span> Onbepaaldheid oneindig min oneindig met breuken <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> Wanneer <strong>de onbepaaldheid oneindig min oneindig optreedt bij het optellen of aftrekken van algebra\u00efsche breuken<\/strong> , moeten we eerst de breuken optellen of aftrekken en vervolgens de limiet berekenen.<\/p>\n<\/div>\n<p> Laten we eens kijken hoe we de onbepaaldheid oneindig min oneindig in een functie met breuken kunnen berekenen door stap voor stap een voorbeeld op te lossen:<\/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-c58eb86af2eb0393a802fc7a29f8a453_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1} - \\frac{x}{3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"152\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Laten we eerst 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-b3a2cbbfec28f9de05668b90e9ee65f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\left(  \\frac{x^2}{x-1} - \\frac{x}{3}\\right) = \\frac{(+\\infty)^2}{(+\\infty)-1} - \\frac{+\\infty}{3} = \\bm{+\\infty - \\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"410\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Maar we verkrijgen de onbepaaldheid \u221e-\u221e.<\/p>\n<p> Dus eerst moeten we breuken aftrekken. Om dit te doen, reduceren we breuken tot een gemeenschappelijke noemer, dat wil zeggen, we vermenigvuldigen de teller en de noemer van de ene breuk met de noemer van de andere:<\/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-68e489c5833478cb20929ea07ae2971d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to +\\infty} \\left( \\frac{x^2}{x-1}-\\frac{x}{3}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to +\\infty}\\left(\\frac{x^2 \\cdot 3}{(x-1)\\cdot 3}- \\frac{x\\cdot (x-1)}{3\\cdot (x-1)} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x \\to +\\infty} \\left( \\frac{3x^2 }{3(x-1)}- \\frac{x^2-x}{3(x-1)}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> En nu beide breuken dezelfde noemer hebben, kunnen we ze combineren tot \u00e9\u00e9n 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-1e5345a6d68ae0cdda543b81f89daa62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{3x^2 -(x^2-x)}{3(x-1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"163\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> We werken in de teller en de noemer:<\/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-31cbae0091a641d74250fae5758b3116_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}  \\frac{3x^2 -x^2+x}{3x-3} =  \\lim_{x \\to +\\infty}  \\frac{2x^2+x}{3x-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"284\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> En tot slot berekenen we de limiet opnieuw:<\/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-6ef29c026035a5353b2bada5bc0d9ff9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{2x^2+x}{3x-3}=\\frac{+\\infty}{+\\infty}=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"225\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p> In dit geval geeft de oneindige onbepaaldheid tussen oneindigheid +\u221e omdat de graad van de teller groter is dan de graad van de noemer.<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <a href=\"https:\/\/mathority.org\/nl\/onbepaaldheid-oneindig-tussen-oneindig-%e2%88%9e-%e2%88%9e\/\"><span style=\"text-decoration: underline;\">wat is oneindig tussen oneindig?<\/span><\/a> <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"indeterminacion-infinito-menos-infinito-con-raices\"><\/span> onbepaaldheid oneindig min oneindig met wortels <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div style=\"background:linear-gradient(to bottom, #FFFFFF 0%, #FFE0B2 100%); padding-top: 23px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px dashed #FF9B28; border-radius:20px; margin-bottom:30px\">\n<p style=\"text-align:left\"> Wanneer <strong>onbepaaldheid oneindig min oneindig optreedt bij radicale optelling of aftrekking<\/strong> , moeten we eerst de functie vermenigvuldigen en delen door de geconjugeerde radicale uitdrukking, en vervolgens de limiet oplossen.<\/p>\n<\/div>\n<p> We zullen zien hoe we de onbepaaldheid oneindig min oneindig in een irrationele functie kunnen oplossen met behulp van een stapsgewijs voorbeeld:<\/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-e093b62c357684fe8a8818df58d7b99a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"165\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> Laten we eerst proberen de limiet van de functie met radicalen op te lossen:<\/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-4459c2b6c968344878499cfbb30adda4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)=+\\infty-\\sqrt{(+\\infty)^2}=\\bm{+\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"409\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p> We verkrijgen echter de onbepaalde vorm \u221e-\u221e. Om te weten hoeveel onbepaaldheid oneindig minus oneindig is, moeten we dus de uitgelegde procedure toepassen.<\/p>\n<p> Omdat de functie radicalen heeft, vermenigvuldigen en delen we de gehele functie door de geconjugeerde irrationele uitdrukking:<\/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-f10d91882a0f8dcca86fbb8dda7da7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(x-\\sqrt{x^2-5}\\right)= \\lim_{x \\to +\\infty}\\frac{\\left(x-\\sqrt{x^2-5}\\right)\\cdot\\left(x+\\sqrt{x^2-5}\\right)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"488\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> De algebra\u00efsche uitdrukking van de teller komt overeen met de opmerkelijke identiteit van het product van een som door een verschil. Daarom kunnen we de uitdrukking vereenvoudigen:<\/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-b00f177bdb579dabf9dc589e387344cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\cfrac{\\left(x-\\sqrt{x^2-5}\\right) \\cdot \\left(x + \\sqrt{x^2-5}\\right)}{ x + \\sqrt{x^2-5}}= \\lim_{x \\to +\\infty} \\cfrac{x^2- \\left( \\sqrt{x^2-5}\\right)^2}{ x + \\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"505\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> We vereenvoudigen nu de wortel van de limiet, omdat deze in het kwadraat 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-d5c798f099ef1c56a50526e7fba8c99c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty}\\frac{x^2-(x^2-5)}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> We werken met de teller 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-392ae211b16ad803eb70cc4993a0c7e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{x^2- x^2+5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -16px;\"><\/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-be954eaf609b9f98c6dc984758599b5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"146\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> En ten slotte voeren we de berekening van de limiet opnieuw uit:<\/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-c29edfa5eba2fe54e369c3d963d11a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to +\\infty} \\frac{5}{x+\\sqrt{x^2-5}}=\\frac{5}{+\\infty+\\sqrt{(+\\infty)^2}}=\\frac{5}{+\\infty}=\\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"391\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> Het resultaat van de limiet is daarom 0, omdat elk getal gedeeld door oneindig gelijk is aan nul. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-indeterminacion-infinito-menos-infinito\"><\/span> Opgeloste oneindige minus oneindige onbepaaldheidsproblemen<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Los de volgende limiet op als x plus oneindig nadert: <\/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-bac36c53d4e34c6e9972009b34a64c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(7x^2-2x^3)\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"134\" style=\"vertical-align: -13px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> In deze limiet is de term van de hoogste orde van de derde graad, dus concentreren we ons op de oneindigheid die uit deze term wordt verkregen. <\/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-eee2b50312f5fd4225c85387c311eec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}(+7x^2-2x^3)=+\\infty^2-\\infty^3=+\\infty-\\infty=\\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"408\" style=\"vertical-align: -13px;\"><\/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> Bereken de limiet van de volgende polynoomfunctie als x de negatieve oneindigheid nadert: <\/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-12b903d03d726c28b625fe3f5ba4b3c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3-9x^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"148\" style=\"vertical-align: -12px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Negatieve oneindigheid in de kubus blijft negatief, maar in het kwadraat wordt het positief. later Hoewel hun tekens worden gewijzigd door de co\u00ebffici\u00ebnten ervoor:<\/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-172f2927f65e61079b13abd02234f1c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\\\[3ex]=-5(-\\infty)^3-9(-\\infty)^2=\\\\[3ex]=-5\\cdot (-\\infty)-9\\cdot \\infty=\\\\[3ex]=+\\infty-\\infty\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"153\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vervolgens wordt de onbepaalde vorm oneindig min oneindig gedefinieerd door de hoogste term (-5x <sup>3<\/sup> ), waaruit we positieve oneindigheid verkrijgen: <\/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-dfd806b31a588234442f48fa5ae8b751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to -\\infty}(-5x^3+x^2)=\\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"194\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 3<\/h3>\n<p> Bepaal de limiet tot oneindig 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-f0a8401379d90875626b1fbd3714fd01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"160\" style=\"vertical-align: -17px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst proberen we de limiet te berekenen door oneindig in de functie te vervangen:<\/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-43df032007d76e00f2f7366e05f9e697_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(\\frac{x^3+1}{x-1}-\\frac{x}{4}\\right)=\\frac{(+\\infty)^3+1}{+\\infty-1}-\\frac{+\\infty}{4} = \\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"425\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we eindigen met de onbepaaldheid \u221e \u2013 \u221e. Daarom herleiden we de breuken tot een gemeenschappelijke noemer:<\/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-7e2820674bc86d085f6deec7fdf9adf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim\\limits_{x \\to +\\infty} \\left(\\frac{x^3+1}{x-1}-\\frac{x}{4} \\right)=\\\\[5ex]\\displaystyle = \\lim_{x\\to +\\infty}\\left(\\frac{(x^3+1)\\cdot4}{(x-1)\\cdot4}-\\frac{x\\cdot(x-1)}{4\\cdot (x-1)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"188\" width=\"302\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En aangezien beide breuken nu dezelfde noemer hebben, kunnen we ze combineren tot \u00e9\u00e9n 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-93a00027be74b1e60c7ee8537ebe5d9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(\\frac{4x^3+4}{4x-4}-\\frac{x^2-x}{4x-4}\\right)=\\lim_{x\\to +\\infty}\\frac{4x^3+4-(x^2-x)}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"429\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We maken de haakjes van 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-c7de3ead5b3a5f8bd2ae8d767da693b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\frac{4x^3+4-x^2+x}{4x-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"180\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte bepalen 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-7ffb73fbf26fd2b625e43872a9c10ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{4x^3+4-x^2+x}{4x-4}=\\frac{4(+\\infty)^3}{4(+\\infty)}=\\frac{+\\infty}{+\\infty} = \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"384\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval geeft de onbepaaldheid \u221e\/\u221e +\u221e omdat de graad van de teller groter is dan de graad van de noemer.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 4<\/h3>\n<p> Los de limiet van de volgende fractionele functie op als x 0 nadert: <\/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-e783bc22baa422d4b537fae4628fb4a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"165\" style=\"vertical-align: -17px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We proberen eerst zoals gewoonlijk 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-207bc08385f430f0f8c49ac34a10f811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\frac{-3\\cdot0-2}{0^4}-\\frac{5}{0^2}=\\frac{-2}{0}-\\frac{5}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"477\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we krijgen de onbepaalde vorm \u221e-\u221e. Daarom moeten we de fracties van de functie terugbrengen tot een gemeenschappelijke noemer.<\/p>\n<p class=\"has-text-align-left\"> In dit geval is x <sup>4<\/sup> een veelvoud van x <sup>2<\/sup> , dus door simpelweg de teller en de noemer van de tweede breuk te vermenigvuldigen met x <sup>2<\/sup> krijgen we dat beide breuken dezelfde noemer hebben:<\/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-876115dc1fb49e81373d70be5fdcfb5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5}{x^2}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5\\cdot x^2}{x^2\\cdot x^2} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We kunnen nu de twee breuken aftrekken:<\/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-bf56e81e075d9ac498e9df87a94a675f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0}\\left(\\frac{-3x-2}{x^4}-\\frac{5x^2}{x^4}\\right)=\\lim_{x\\to 0}\\frac{-3x-2-5x^2 }{x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"346\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We proberen de limiet opnieuw op te lossen:<\/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-b231cf80ccb03d1287c1aab47769bc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0}  \\cfrac{-3x-2-5x^2 }{x^4} =\\cfrac{-3\\cdot 0-2-5\\cdot 0^2}{0^4}=\\frac{-2}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"370\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we eindigen met de onbepaaldheid van een constante gedeeld door nul. Het is daarom noodzakelijk om de laterale grenzen van de functie 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-c4ced459b1e0da92f03d9d9515b6ea68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{-}} \\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/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-239f065e0fe7bb4055e63a8477c030f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 0^{+}}\\frac{-3x-2-5x^2}{x^4}=\\frac{-2}{+0}=-\\infty\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"262\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluderend: aangezien de twee laterale grenzen van de functie op het punt x=0 -\u221e opleveren, is de oplossing van de limiet -\u221e: <\/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-30ab5fa39e1b25568d55de0cc4267dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x \\to 0^-}f(x)=\\lim_{x \\to 0^+}f(x)=-\\infty\\ \\longrightarrow \\  \\lim_{x \\to 0}f(x)= \\bm{-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"401\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 5<\/h3>\n<p> Los de limiet tot oneindig op van de volgende functie met wortels: <\/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-5fb2be8c217ffddadf1b3d9d55f100c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"182\" style=\"vertical-align: -13px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als we de limiet proberen op te lossen, verkrijgen we de onbepaaldheid oneindig min oneindig:<\/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-8a1a6b3ff08a703378b8cfb1b5e6532c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(4x^2-\\sqrt{x^4+1}\\right)=4(+\\infty)^2-\\sqrt{(+\\infty)^4}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"456\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Omdat er radicalen in de functie zitten, moeten we deze daarom vermenigvuldigen en delen door de geconjugeerde radicaaluitdrukking:<\/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-3c4cdc9585a792800b8c903745ecc7c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\left(4x^2-\\sqrt{x^4+1} \\right)=\\lim_{x \\to +\\infty}\\frac{\\left(4x^2-\\sqrt{x^4+1}\\right)\\cdot\\left(4x^2+\\sqrt{x^4+1}\\right)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"538\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In de teller hebben we het opmerkelijke product van een som met een verschil, dat gelijk is aan het verschil van de kwadraten. Nog:<\/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-1aab32f1a28189a4ce96f3816f11a02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(4x^2\\right)^2-\\left(\\sqrt{x^4+1}\\right)^2}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"216\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vereenvoudigen de radicaal tot het kwadraat:<\/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-6d86adc198c4fb2cd1d99c94e5b8430e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\bigl(4x^2\\bigr)^2-(x^4+1)}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"186\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We werken in 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-5c07c403048b4d3e40a8034333ff069c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{16x^4-x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/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-c138b064a8fa3142cb2d50782807ebb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"163\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En uiteindelijk vinden 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-cdb8845be6c640f0370961c3a52598d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{15x^4-1}{4x^2+\\sqrt{x^4+1}}=\\frac{15(+\\infty)^4}{4(+\\infty)^2+\\sqrt{(+\\infty)^4}}=\\frac{+\\infty}{+\\infty}= \\bm{+\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"460\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval is de oneindige onbepaaldheid gedeeld door oneindig oneindiger omdat de graad van de teller groter is dan de graad van de noemer (bedenk dat de vierkantswortel de graad met twee reduceert:<\/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-ded55d5413ed7bccc29e8228df205f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x^4} = x^{4\/2} = x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -1px;\"><\/p>\n<p> ).<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 6<\/h3>\n<p> Los de limiet op als x de oneindigheid nadert van de volgende irrationele 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-7d9f21f0159778cdb1f0710e1a9e0023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"214\" style=\"vertical-align: -13px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst proberen we de limiet zoals gewoonlijk 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-5419056e772f9d11884cae7e315ca947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\left(2x-1-\\sqrt{4x^2+1}\\right)=2(+\\infty)-\\sqrt{4(+\\infty)^2}=\\bm{+\\infty -\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"489\" style=\"vertical-align: -13px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar het geeft ons als resultaat de onbepaaldheid van het verschil van oneindigheden. Omdat de functie wortels heeft, moeten we de uitdrukking daarom vermenigvuldigen en delen door het geconjugeerde radicaal:<\/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-bde8a1f86cf7be80170b9595b5a822df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1-\\sqrt{4x^2+1}\\right)\\cdot\\left(2x-1+\\sqrt{4x^2+1}\\right)}{2x-1 +\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"393\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We groeperen de opmerkelijke gelijkheid van de teller 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-e074e8c7841e0951ae03d6dfd2bfd1b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(\\sqrt{4x^2+1}\\right)^2}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We lossen de vierkantswortel 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-17beafd120a7fc185e1499671fb4421a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{\\left(2x-1\\right)^2-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"218\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We lossen de opmerkelijke identiteit van het kwadraat van een verschil 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-4a34cb3941c92a785c11c50ecaa1e438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-\\left(4x^2+1\\right)}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"245\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We werken in 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-1a2e8d86f22087e775650d36bf78e719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{4x^2+1-4x-4x^2-1}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"228\" style=\"vertical-align: -16px;\"><\/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-2f25890bccb1eaa4c7aa7338f3a25f6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"195\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte berekenen we de waarde van de limiet op oneindig:<\/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-6986ded778a6220e3ad9d6c6bf873451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to +\\infty} \\cfrac{-4x }{2x-1 +\\sqrt{4x^2+1} } = \\cfrac{-4(+\\infty) }{2(+\\infty)+\\sqrt{4(+\\infty)^2} } = \\cfrac{-\\infty}{+\\infty} =\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"458\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ook al staat er een x kwadraat in de noemer, de graad ervan is eigenlijk 1 omdat deze binnen een wortel staat:<\/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-0decc88d206f476d332becb025b8eeaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2} =\\sqrt{4}\\cdot \\sqrt{x^2} = \\sqrt{4}\\cdot x^{2\/2} =\\sqrt{4} x^1=\\sqrt{4}x .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"351\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daarom is het resultaat van de onbepaaldheid -\u221e\/+\u221e de deling van de co\u00ebffici\u00ebnten van de x van hogere graad, aangezien de graad van de teller hetzelfde is als de graad van de noemer.<\/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-8eb19af7ca51c14245db81bd6781b881_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to +\\infty}\\frac{-4x}{2x-1+\\sqrt{4x^2+1} }=\\frac{-\\infty}{+\\infty}=\\frac{-4}{2+\\sqrt{4}}=\\frac{-4}{2+2}=\\frac{-4}{4}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"499\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Merk op dat er twee eerstegraadstermen in de noemer staan<\/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-c973910499b6b5a4828e213dc33f948d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl(2x\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"25\" style=\"vertical-align: -7px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c623cb17f27418239e3fcf7c2ec09946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}\\bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"46\" style=\"vertical-align: -7px;\"><\/p>\n<p> , om de onbepaaldheid -\u221e\/+\u221e op te lossen is het noodzakelijk om alle co\u00ebffici\u00ebnten van de eerstegraadstermen te nemen, dat wil zeggen de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da4556c0a02b580047678d308649edf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> en de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65ddaa07508d3929b6969a5e4e6baddf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\"><\/p>\n<p> van <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e8a851efdbfbb4531c82837d5a61edd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{4x^2}.\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 7<\/h3>\n<p> Bereken de limiet als x 1 van de volgende functie met breuken benadert: <\/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-480bb119c1303a7afa394d812b0e7602_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"173\" style=\"vertical-align: -17px;\"><\/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 oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als we proberen de limiet te bereiken, krijgen we de onbepaalde limiet van oneindig min oneindig:<\/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-d11d45ea6681f3645773f6e0df8cce9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3}\\right)=\\frac{1}{1-1}--\\frac{3}{1-1^3}=\\frac{1}{0}-\\frac{3}{0}=\\bm{\\infty-\\infty}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"480\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We moeten daarom de breuken herleiden tot een gemeenschappelijke noemer, of met andere woorden, we moeten de teller en de noemer van de ene breuk vermenigvuldigen met de noemer van de andere:<\/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-75bf3ffa177f32711c5509ce5fe5992d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1}{1-x}-\\frac{3}{1-x^3} \\right)=\\\\[5ex]\\displaystyle =\\lim_{x\\to 1}\\left( \\frac{1\\cdot(1-x^3)}{(1-x)\\cdot(1-x^3)}-\\frac{3\\cdot(1-x)}{(1-x^3)\\cdot(1-x)}\\right)=\\\\[5ex]\\displaystyle =\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"186\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En aangezien de twee breuken nu dezelfde noemer hebben, kunnen we ze samenvoegen:<\/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-c381a263e89e5a60ff0e6df9367a8ab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\left(\\frac{1-x^3}{1-x-x^3+x^4}-\\frac{3-3x}{1-x-x^3+x^4}\\right)=\\lim_{x\\to 1}\\frac{1-x^3-(3-3x)}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"517\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wij opereren: <\/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-05279cd25d55f5c50edfb5f82929701b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{1-x^3-3+3x}{1-x-x^3+x^4}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/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-818107141eb339d788408e23078ddda9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1} \\cfrac{-x^3+3x-2}{x^4-x^3-x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we proberen de limiet opnieuw op te lossen:<\/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-9d0a31b51faff7e77e778fba66fdbaa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\frac{-1^3+3\\cdot1-2}{1^4-1^3-1+1}=\\mathbf{\\frac{0}{0}}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"335\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we vinden de onbepaaldheid nul gedeeld door nul. We moeten daarom de teller- en noemerpolynomen in factoren ontbinden:<\/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-e5b8321a511b5e370abe8844bf9624ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-x^3+3x-2}{x^4-x^3-x+1}=\\lim_{x \\to 1}\\frac{-(x-1)^2(x+2)}{(x-1)^2(x^2+x+1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"369\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vereenvoudigen we de breuk door de factor te verwijderen die zich herhaalt in de teller en de noemer:<\/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-ab5629bd2fabeb755da37d3abea335b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-\\cancel{(x-1)^2}(x+2)}{\\cancel{(x-1)^2}(x^2+x+1)}=\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"329\" style=\"vertical-align: -17px;\"><\/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-dbb1676133fe1e33fb4d18078b945959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x \\to 1}\\frac{-(x+2)}{x^2+x+1}=\\frac{-(1+2)}{1^2+1+1}=\\frac{-3}{3}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"316\" style=\"vertical-align: -14px;\"><\/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 hoe je de oneindige minus oneindige (\u221e-\u221e) onbepaaldheid kunt oplossen. Je vindt voorbeelden van deze onbepaaldheid met verschillende soorten functies en daarnaast kun je oefenen met oefeningen die stap voor stap worden opgelost van onbepaaldheid oneindig min oneindig. Het oplossen van onbepaaldheid oneindig min oneindig Wanneer de limiet van &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/onbepaaldheid-oneindig-min-oneindig-%e2%88%9e-%e2%88%9e\/\"> <span class=\"screen-reader-text\">Onbepaaldheid oneindig min oneindig (\u221e-\u221e)<\/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-419","post","type-post","status-publish","format-standard","hentry","category-functielimieten"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Onbepaaldheid oneindig minus oneindig (\u221e-\u221e) - Mathoriteit<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/nl\/onbepaaldheid-oneindig-min-oneindig-\u221e-\u221e\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Onbepaaldheid oneindig minus oneindig (\u221e-\u221e) - Mathoriteit\" \/>\n<meta property=\"og:description\" content=\"In dit artikel leggen we uit hoe je de oneindige minus oneindige (\u221e-\u221e) onbepaaldheid kunt oplossen. 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