{"id":63,"date":"2023-09-17T11:08:30","date_gmt":"2023-09-17T11:08:30","guid":{"rendered":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/"},"modified":"2023-09-17T11:08:30","modified_gmt":"2023-09-17T11:08:30","slug":"trigonometrische-grenzen","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/","title":{"rendered":"Trigonometrische limieten"},"content":{"rendered":"<p>Hier leert u hoe u trigonometrische limieten oplost. Je zult verschillende voorbeelden van limieten van goniometrische functies kunnen zien en zelfs oefenen met opgeloste stapsgewijze oefeningen over goniometrische limieten. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-limites-trigonometricos\"><\/span> Wat zijn trigonometrische limieten?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Trigonometrische limieten zijn limieten die worden berekend op basis van goniometrische functies.<\/strong> Om trigonometrische limieten op te lossen, moet een voorbereidende procedure worden toegepast, omdat deze doorgaans aanleiding geven tot onbepaaldheid.<\/p>\n<p> Bovendien bestaan er geen oneindige limieten voor trigonometrische functies, omdat het periodieke functies zijn. Dat wil zeggen dat de grafieken voortdurend periodiek worden herhaald zonder naar een specifieke waarde te neigen. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formulas-de-los-limites-trigonometricos\"><\/span> Trigonometrische limietformules<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Alle trigonometrische limieten worden berekend op basis van de volgende twee formules: <\/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-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" 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>Formule demonstratie<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als we de limiet proberen te berekenen door middel van substitutie, verkrijgen we de nulbepaaldheid tussen nul:<\/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-0ba13ab3640b429e546e97da2a0ab155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=\\frac{\\text{sen}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"193\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar deze trigonometrische formule kan worden gedemonstreerd door waarden te berekenen van de dichtstbijzijnde functie en dichter bij x=0 (hoeken in radialen). <\/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-2a7ed3df9110a97c224bde10980f2682_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"142\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-123\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ccd668acef73b9140a0cbbb9c1d53ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(-1)=\\cfrac{\\text{sen}(-1)}{-1}=0,84147\\\\[3ex]f(-0,1)=\\cfrac{\\text{sen}(-0,1)}{-0,1}=0,99833\\\\[3ex]f(-0,01)=\\cfrac{\\text{sen}(-0,01)}{-0,01}=0,99998\\\\[3ex]f(-0,001)=\\cfrac{\\text{sen}(-0,001)}{-0,001}=0,99999\\end{array}\\\\[14ex]\\vdots\\\\[2ex]\\displaystyle\\lim_{x\\to 0^-}\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"312\" width=\"288\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-66152efc3ce1fa761186a65db677af27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\begin{array}{l}f(1)=\\cfrac{\\text{sen}(1)}{1}=0,84147\\\\[3ex]f(0,1)=\\cfrac{\\text{sen}(0,1)}{0,1}=0,99833\\\\[3ex]f(0,01)=\\cfrac{\\text{sen}(0,01)}{0,01}=0,99998\\\\[3ex]f(0,001)=\\cfrac{\\text{sen}(0,001)}{0,001}=0,99999\\end{array}\\\\[14ex]\\vdots\\\\[2ex]\\displaystyle\\lim_{x\\to 0^+}\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"312\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De twee laterale grenzen van de goniometrische functie geven 1, dus de limiet op het punt x=0 is 1:<\/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-8af649189957b154866097e315f7cb92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}\\displaystyle\\lim_{x\\to 0^-}\\frac{\\text{sen}(x)}{x}=\\lim_{x\\to 0^+}\\frac{\\text{sen}(x)}{x}=1\\\\[3ex]\\color{orange}\\bm{\\downarrow}\\\\[2ex]\\lim_{x\\to 0}\\displaystyle\\frac{\\text{sen}(x)}{x}=1\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"130\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De trigonometrische limiet van de sinus van x gedeeld door x, aangezien x naar 0 neigt, is dus gelijk aan 1.<\/p>\n<p class=\"has-text-align-left\"> Deze formule kan ook voor verschillende hoeken worden toegepast: <\/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-e8b276a8e0f8bf93f3ea2b7d0158adbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(kx)}{kx}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"125\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" 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>Formule demonstratie<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als we proberen de limiet te vinden door directe substitutie, krijgen we de onbepaalde vorm nul tussen nul:<\/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-3e50a5f25f1ca148a4e0107e75e62c43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=}\\frac{1-\\text{cos}(0)}{0}=\\frac{1-1}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"319\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we kunnen de gelijkheid controleren aan de hand van de bovenstaande formule. Om dit te doen, moet je eerst de teller en de noemer van de breuk vermenigvuldigen met 1 plus de cosinus van x:<\/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-40196b4f425393970ff11577ef645dba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\bigl(1-\\text{cos}(x)\\bigr)\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"48\" width=\"235\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We hebben nu een opmerkelijke identiteit in de teller van de breuk, dus we kunnen deze 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-052fdb01d818c3baa3293d4e1927d37c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1^2-\\text{cos}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/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-4b7f2f3a29d5eaebb5f226607e80dbb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vertrekkend van de fundamentele trigonometrische identiteit herschrijven we 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-ff23bded6a6a479ee358e635c74ef2fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(x)+\\text{cos}^2(x)=1 \\ \\longrightarrow \\ \\text{sen}^2(x)=1-\\text{cos}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"381\" style=\"vertical-align: -5px;\"><\/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-df11237bdbf1c1ef5f607f08db54ca91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}^2(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"151\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We kunnen de breuk daarom omzetten in een product van breuken: <\/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-cff26553bbe1117d69b5a11e0371b996_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)\\cdot \\text{sen}(x)}{x\\cdot \\bigl(1+\\text{cos}(x)\\bigr)}\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"151\" style=\"vertical-align: -20px;\"><\/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-ac46678f2030eed0dc15696613ec60ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Met behulp van de eigenschappen van limieten kunnen we de bovenstaande uitdrukking omzetten in een product van limieten:<\/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-94b986a7b575a61eeba306ce22a6a01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"209\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Met behulp van de hierboven gedemonstreerde formule kunnen we de trigonometrische limiet eenvoudig 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-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" 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-6c46c6322634327f17aa601618460fd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"133\" 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-b446be13fe115291c38a7c34c192d571_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{\\text{sen}(x)}{1+\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"112\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte berekenen we de resulterende 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-bee04c527b0609fc39a7729ec6677874_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{\\text{sen}(0)}{1+\\text{cos}(0)}=\\frac{0}{1+1}=\\frac{0}{2}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"248\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Daarom wordt de trigonometrische limietformule geverifieerd:<\/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-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Net als de andere formule kan deze ook voor meerdere hoeken worden gebruikt: <\/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-afd4adcffaaad5d5b5c7063ec3542b5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(kx)}{kx}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"156\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> <strong>Om trigonometrische limieten op te lossen, moeten we daarom rekenkunde gebruiken om de functies te transformeren en soortgelijke uitdrukkingen te verkrijgen.<\/strong> Op deze manier kunnen we een van de twee formules gebruiken en de waarde van de limiet vinden.<\/p>\n<p> Aan de andere kant moeten we soms bepaalde trigonometrische identiteiten toepassen, dus laten we alle onderstaande formules aan u over <\/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>Trigonometrische identiteiten<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Formule die de drie belangrijkste trigonometrische verhoudingen met elkaar verbindt:<\/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-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Fundamentele trigonometrische identiteit:<\/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-92d80771f891319379b2e756c5524aaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}^2(x)+\\text{cos}^2(x)=1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Trigonometrische relaties afgeleid van de fundamentele: <\/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-983ec3f9bdead575a110ab13a3149351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+\\text{tan}^2 (x)=\\cfrac{1}{\\text{cos}^2(x)}=\\text{sec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"245\" 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-7b04254cbdd6156ce5fd5449f5234a9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1+\\text{cot}^2 (x)=\\cfrac{1}{\\text{sen}2(x)}=\\text{cosec}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"262\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tegenovergestelde hoeken: <\/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-a0a5345d1ad85390cacfc38e99beb548_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(-x)=-\\text{sen}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/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-b7ef3e0d227838cf04c0f7413d1e07f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(-x)=\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -5px;\"><\/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-507f1aa7df63922130ea766d03aaf91a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(-x)=-\\text{tan}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Som van twee hoeken: <\/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-58668a0aaa63a0e4c39b859619d2444a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(x+y)=\\text{sen}(x)\\text{cos}(y)+\\text{cos}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/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-46aaa0aea1219b24ef354afcc8a15953_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(x+y)=\\text{cos}(x)\\text{cos}(y)-\\text{sen}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"314\" style=\"vertical-align: -5px;\"><\/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-c5c001a17b792285beacf6cf91f93033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x+y)=\\cfrac{\\text{tan}(x)+\\text{tan}(y)}{1-\\text{tan}(x)\\text{tan}(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"235\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Verschil van twee hoeken: <\/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-02420c87f520da509e0193dab4798f55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(x-y) = \\text{sen}(x)\\text{cos}(y)-\\text{cos}(x)\\text{sen}(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"315\" style=\"vertical-align: -5px;\"><\/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-9d5d84d5fa7db15e90131596953bedb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(x-y) = \\text{cos}(x)\\text{cos}(y)+ \\text{sen}(x) sen(y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"317\" style=\"vertical-align: -5px;\"><\/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-b5d01deaacd8e294bcfd6b6284231fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x-y)=\\cfrac{\\text{tan}(x)-\\text{tan}(y)}{1+\\text{tan}(x)\\text{tan}(y)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"235\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dubbele hoek: <\/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-c135a8fb824883a8b8f9ff27a737a9d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(2x) = 2\\text{sen}(x)\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -5px;\"><\/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-91843029bf168eab0615f3bb849f2dd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{cos}(2x) =\\text{cos}^2(x)-\\text{sen}^2(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"213\" style=\"vertical-align: -5px;\"><\/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-0cc8aed863858c3052e1dae8bdcdb377_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(2x) =\\cfrac{2\\text{tan}(x)}{1-\\text{tan}^2(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"172\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Halve hoek: <\/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-ccae2dc8b2bc812d68f9361538ebaf4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1-\\text{cos}(x)}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"201\" style=\"vertical-align: -13px;\"><\/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-6f87b3d54e5b0d7527bf38b2a7a71928_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1+\\text{cos}(x)}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"200\" style=\"vertical-align: -13px;\"><\/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-1126d8d443e0285d4ecc510a119b393d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\text{tan}\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\frac{1-\\text{cos}(x)}{1+\\text{cos}(x)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"202\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Optellen en aftrekken van sinus en cosinus: <\/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-457b2949084f43244b619fd965e403f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)+\\text{sen}(y)=2\\text{sen}\\left(\\frac{x+y}{2} \\right)\\text{cos}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"347\" 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-4ff2373c8559ef4bebc00a31c7c8f2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)-\\text{sen}(y)=2\\text{cos}\\left(\\frac{x+y}{2} \\right)\\text{sen}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"347\" 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-625d75c4be2e5bbaca73e2a3f1e1980b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)+\\text{cos}(y)=2\\text{cos}\\left(\\frac{x+y}{2} \\right)\\text{cos}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"344\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0a498d3701c9c87123c269c81d266d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)-\\text{cos}(y)=-2\\text{sen}\\left(\\frac{x+y}{2} \\right)\\text{sen}\\left(\\frac{x-y}{2} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"360\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Product van sinussen en cosinussen: <\/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-78ab2bb9d2bc291a1f7e4c9e329d893e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)\\cdot \\text{sen}(y)=\\frac{1}{2}\\Bigl[\\text{cos}(x-y)-\\text{cos}(x+y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"338\" 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-40daecfe989acfa36adb6772d193d027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{cos}(x)\\cdot \\text{cos}(y)=\\frac{1}{2}\\Bigl[\\text{cos}(x+y)+\\text{cos}(x-y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"336\" 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-b01d45ae9c7b57d25e2bd1bdfea9dba9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{sen}(x)\\cdot \\text{cos}(y)=\\frac{1}{2}\\Bigl[\\text{sen}(x+y)+\\text{sen}(x-y)\\Bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"339\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Om u precies te laten zien hoe trigonometrische limieten worden berekend, hebben we hieronder een stapsgewijs voorbeeld samengesteld.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-limite-trigonometrico\"><\/span> Voorbeeld van trigonometrische limiet<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Laten we eens kijken hoe een trigonometrische limiet wordt opgelost aan de hand van het volgende 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-83fe05cfbae51406227f863405374405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"83\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Als we proberen de trigonometrische limiet te berekenen, verkrijgen we de onbepaaldheid van nul tussen nul:<\/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-29fd7943ebb7d7c4ecc7886207c4a1cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}=\\frac{\\text{tan}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"195\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Zie:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/nl\/nul-tussen-nul-0-0-onbepaaldheid\/\">nulgrenzen tussen nul<\/a><\/span><\/p>\n<p> Het is daarom noodzakelijk om de trigonometrische functie te transformeren om de limiet op te lossen. De tangensfunctie is gelijk aan de sinus gedeeld door de cosinus, dus:<\/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-dbf6d65fa67f0a2161bd99ee7431f015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"124\" 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-24793b3f48b399e9fd64b2eb6758f0c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)}{x}=\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"195\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> We kunnen de functie nu als een product uitdrukken 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-fe877e223e062371ef4aa551372cfa69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\frac{\\displaystyle\\frac{a}{b}}{\\displaystyle\\frac{c}{d}}=\\frac{a\\cdot d}{b\\cdot c}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"73\" style=\"vertical-align: -30px;\"><\/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-dd10c321c3b7dc40698b318c7187a3c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{\\displaystyle\\frac{x}{1}}=\\lim_{x\\to 0}{\\frac{\\text{sen}(x)\\cdot 1}{\\text{cos}(x) \\cdot x}=\\\\[6ex]\\displaystyle =\\lim_{x\\to 0}{\\frac{\\text{sen}(x)}{x\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot \\frac{1}{\\text{cos}(x)}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"145\" width=\"282\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Met behulp van de eigenschappen van limieten kunnen we de limiet van twee vermenigvuldigde functies omzetten naar het product van twee limieten:<\/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-38f278cf96ac97997db5ffe530037582_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\frac{1}{\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"350\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Zoals we hierboven hebben laten zien, geeft de eerste trigonometrische limiet 1:<\/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-14300dcf4010d732b8568b9b4460b5e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"116\" 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-8ac5a01bca48ac7a961a99be694dcd8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=1\\cdot\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"413\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Voer dus gewoon de volgende berekening 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-24b950800435e32f07649e25afd6d68e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lim_{x\\to 0}\\frac{1}{\\text{cos}(x)}=\\frac{1}{\\text{cos}(0)}=\\frac{1}{1}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"225\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-limites-trigonometricos\"><\/span> Opgeloste oefeningen over trigonometrische limieten<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Los de volgende trigonometrische 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-730128e3fffaf36349ed1c2db19d8796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"91\" 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 de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst proberen we de trigonometrische limiet te berekenen door directe evaluatie:<\/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-b46144b4ff38b63a3c428b5aa60ffb5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}=\\frac{\\text{sen}(4\\cdot 0)}{2\\cdot 0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"224\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar we krijgen nul boven nul onbepaaldheid. We moeten dus transformaties op de functie toepassen.<\/p>\n<p class=\"has-text-align-left\"> Eerst laten we de x in de noemer staan door het volgende te doen:<\/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-66cfa67d319a268be5ac3c6eaf733240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{2x}=\\lim_{x\\to 0}\\frac{1}{2}\\cdot\\frac{\\text{sen}(4x)}{x}=\\frac{1}{2}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"375\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vermenigvuldigen en delen we de breuk door 4 om een uitdrukking te verkrijgen waarmee de eerste formule voor trigonometrische limieten kan worden toegepast:<\/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-caa05571515d0dbc716d6e8cb6b0be0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\frac{1}{2}\\lim_{x\\to 0}\\frac{\\text{sen}(4x)\\cdot 4}{x\\cdot 4}=\\frac{1}{2}\\cdot 4 \\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}=2\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"418\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten slotte passen we de formule toe die we aan het begin hebben gezien en lossen we de trigonometrische 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-e8b276a8e0f8bf93f3ea2b7d0158adbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(kx)}{kx}=1\" 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-e21a4ad74086fda398299e2d83c9a052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2\\lim_{x\\to 0}\\frac{\\text{sen}(4x)}{4x}=2\\cdot 1=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"190\" 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 2<\/h3>\n<p> Bereken de volgende trigonometrische 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-8b6eb2bc6fa65e96cfe55e695b93b2cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"153\" 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 de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst proberen we de trigonometrische limiet te vinden:<\/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-2e4a361830bf074dfb37219d1288c315_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}=\\frac{\\text{sen}(0)+\\text{tan}(0)}{0}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"334\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Maar de onbepaalde vorm nul komt overeen met nul is bereikt.<\/p>\n<p class=\"has-text-align-left\"> Vervolgens zetten we de raaklijn om in een quoti\u00ebnt van de sinus en de cosinus:<\/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-e8dbbefdca880d5806977d6b13b473b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)+\\text{tan}(x)}{x}=\\lim_{x\\to 0}\\frac{\\displaystyle\\text{sen}(x)+\\frac{\\text{sen}(x)}{\\text{cos}(x)}}{x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"335\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We vermenigvuldigen en delen door de cosinus van x:<\/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-53452abaa54f3c7e46b75965500221ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\left(\\displaystyle\\text{sen}(x)+\\frac{\\text{sen}(x)}{\\text{cos}(x)}\\right)\\cdot\\text{cos}(x)}{x\\cdot\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)\\text{cos}(x)+\\text{sen}(x)}{x\\cdot\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"469\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We nemen een gemeenschappelijke factor in de teller en scheiden de trigonometrische limiet in twee\u00ebn:<\/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-eeaa601256f134072260480b64210950_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)(\\text{cos}(x)+1)}{x\\cdot\\text{cos}(x)}=\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{cos}(x)+1}{\\text{cos}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"408\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En tenslotte vinden we het resultaat van de trigonometrische 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-7259ba5ebcb847c0953947ca2fb1d219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{cos}(x)+1}{\\text{cos}(x)}=1\\cdot\\frac{\\text{cos}(0)+1}{\\text{cos}(0)} =\\frac{1+1}{1}=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"435\" style=\"vertical-align: -17px;\"><\/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> Los de limiet van de volgende trigonometrische functie op als x nul 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-f9da1c65931e93b840821e76bc20d629_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\text{tan}(x)-\\text{sen}{(x)}}{3x\\cdot\\text{tan}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"153\" 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 de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Door de directe berekening uit te voeren, verkrijgen we de onbepaalde limiet 0 tussen 0:<\/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-67e20b7fd699b38122cab6a801cc5655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}}\\frac{\\text{tan}(x)-\\text{sen}(x)}{3x\\cdot\\text{tan}(x)}=\\frac{\\text{tan}(0)-\\text{sen}(0)}{3\\cdot 0\\cdot\\text{tan}(0)}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"334\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We zullen dus de limiet vereenvoudigen door elke term te delen door de tangens van x:<\/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-d9036c709f0cf05a0e1d3e53a1f81af8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle\\frac{\\text{tan}(x)}{\\text{tan}(x)}-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{\\displaystyle\\frac{3x\\cdot\\text{tan}(x)}{\\text{tan}(x)}}=\\lim_{x\\to 0}\\frac{\\displaystyle 1-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"305\" style=\"vertical-align: -40px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten tweede kunnen we uit de fundamentele trigonometrische identiteit afleiden dat de breuk van de teller equivalent is aan de cosinus van x: <\/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-2c4733e791e3ea6006f69e25c3db9f99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{tan}(x)=\\cfrac{\\text{sen}(x)}{\\text{cos}(x)}\\ \\longrightarrow \\ \\text{cos}(x)=\\cfrac{\\text{sen}(x)}{\\text{tan}(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"297\" 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-061c6b3972071af7e6227fad37ec4019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{\\displaystyle 1-\\frac{\\text{sen}(x)}{\\text{tan}(x)}}{3x}=\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{3x}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"255\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En door de tweede formule toe te passen die wordt gedemonstreerd in de theorie van trigonometrische limieten, kunnen we de limiet gemakkelijk oplossen: <\/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-51835176b411ca1ec4f37835a83685fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"146\" 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-b1ee20bc86ad7559fcda3d6bad3c9b27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{3x}=\\lim_{x\\to 0}\\frac{1}{3}\\cdot \\frac{1-\\text{cos}(x)}{x}=\\\\[4ex]\\displaystyle =\\frac{1}{3}\\lim_{x\\to 0}\\frac{1-\\text{cos}(x)}{x}=\\frac{1}{3}\\cdot 0=\\bm{0}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/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> Bepaal de oplossing van de volgende trigonometrische limiet op het punt x=0: <\/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-a7cbd12a8e0f0416e55baa4799395661_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"196\" 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 de oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als we de limiet proberen op te lossen, vinden we de onbepaalde vorm 0\/0:<\/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-0d1052ddde97caedf5e563febc26fad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}=\\frac{2\\text{sen}(0)\\text{cos}(0)\\text{sen}(5\\cdot 0)}{0^2}=\\frac{0}{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"432\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De algebra\u00efsche uitdrukking voor de teller kan worden herschreven met behulp van de trigonometrische identiteit van de sinus van een dubbele hoek: <\/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-2a7acdc1773c3d7fd430328604cee7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{sen}(2x)=2\\text{sen}(x)\\text{cos}(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"185\" style=\"vertical-align: -5px;\"><\/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-4426a183bebf1739384bda14bcd59dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\lim_{x\\to 0}\\frac{2\\text{sen}(x)\\text{cos}(x)\\text{sen}(5x)}{x^2}=\\lim_{x\\to 0}\\frac{\\text{sen}(2x)\\text{sen}(5x)}{x^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"371\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Laten we nu de limiet van de trigonometrische functie in een product scheiden:<\/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-fc650634075435b782f1e7b921b77c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(2x)\\cdot \\text{sen}(5x)}{x\\cdot x}=\\\\[4ex]\\displaystyle =\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\frac{\\text{sen}(5x)}{x}=\\\\[4ex]\\displaystyle =\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{x}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"163\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte lossen we de trigonometrische limiet op door de eigenschappen van limieten 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-7c26ba3032828541e69e4bd976ac4f96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle\\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{x}\\cdot\\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{x}=\\\\[4ex]\\displaystyle =2\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(2x)}{2x}\\cdot 5\\cdot \\lim_{x\\to 0}\\frac{\\text{sen}(5x)}{5x}=\\\\[4ex]\\displaystyle =2\\cdot 1\\cdot 5\\cdot 1=\\bm{10}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"141\" width=\"278\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Hier leert u hoe u trigonometrische limieten oplost. Je zult verschillende voorbeelden van limieten van goniometrische functies kunnen zien en zelfs oefenen met opgeloste stapsgewijze oefeningen over goniometrische limieten. Wat zijn trigonometrische limieten? Trigonometrische limieten zijn limieten die worden berekend op basis van goniometrische functies. Om trigonometrische limieten op te lossen, moet een voorbereidende procedure &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/\"> <span class=\"screen-reader-text\">Trigonometrische limieten<\/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":[61],"tags":[],"class_list":["post-63","post","type-post","status-publish","format-standard","hentry","category-trigonometrie"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Trigonometrische limieten -<\/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\/trigonometrische-grenzen\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trigonometrische limieten -\" \/>\n<meta property=\"og:description\" content=\"Hier leert u hoe u trigonometrische limieten oplost. Je zult verschillende voorbeelden van limieten van goniometrische functies kunnen zien en zelfs oefenen met opgeloste stapsgewijze oefeningen over goniometrische limieten. Wat zijn trigonometrische limieten? Trigonometrische limieten zijn limieten die worden berekend op basis van goniometrische functies. Om trigonometrische limieten op te lossen, moet een voorbereidende procedure &hellip; Trigonometrische limieten Lees meer &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-17T11:08:30+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14300dcf4010d732b8568b9b4460b5e0_l3.png\" \/>\n<meta name=\"author\" content=\"Redactioneel Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Geschreven door\" \/>\n\t<meta name=\"twitter:data1\" content=\"Redactioneel Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Geschatte leestijd\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minuten\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/\",\"url\":\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/\",\"name\":\"Trigonometrische limieten -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/nl\/#website\"},\"datePublished\":\"2023-09-17T11:08:30+00:00\",\"dateModified\":\"2023-09-17T11:08:30+00:00\",\"author\":{\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\"},\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/#breadcrumb\"},\"inLanguage\":\"nl-NL\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/nl\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Trigonometrische limieten\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/nl\/#website\",\"url\":\"https:\/\/mathority.org\/nl\/\",\"name\":\"\",\"description\":\"Waar nieuwsgierigheid en berekening elkaar ontmoeten!\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/nl\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"nl-NL\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64\",\"name\":\"Redactioneel Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"nl-NL\",\"@id\":\"https:\/\/mathority.org\/nl\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Redactioneel Team\"},\"sameAs\":[\"http:\/\/mathority.org\/nl\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Trigonometrische limieten -","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/","og_locale":"nl_NL","og_type":"article","og_title":"Trigonometrische limieten -","og_description":"Hier leert u hoe u trigonometrische limieten oplost. Je zult verschillende voorbeelden van limieten van goniometrische functies kunnen zien en zelfs oefenen met opgeloste stapsgewijze oefeningen over goniometrische limieten. Wat zijn trigonometrische limieten? Trigonometrische limieten zijn limieten die worden berekend op basis van goniometrische functies. Om trigonometrische limieten op te lossen, moet een voorbereidende procedure &hellip; Trigonometrische limieten Lees meer &raquo;","og_url":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/","article_published_time":"2023-09-17T11:08:30+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14300dcf4010d732b8568b9b4460b5e0_l3.png"}],"author":"Redactioneel Team","twitter_card":"summary_large_image","twitter_misc":{"Geschreven door":"Redactioneel Team","Geschatte leestijd":"5 minuten"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/","url":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/","name":"Trigonometrische limieten -","isPartOf":{"@id":"https:\/\/mathority.org\/nl\/#website"},"datePublished":"2023-09-17T11:08:30+00:00","dateModified":"2023-09-17T11:08:30+00:00","author":{"@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64"},"breadcrumb":{"@id":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/#breadcrumb"},"inLanguage":"nl-NL","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/nl\/trigonometrische-grenzen\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/nl\/"},{"@type":"ListItem","position":2,"name":"Trigonometrische limieten"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/nl\/#website","url":"https:\/\/mathority.org\/nl\/","name":"","description":"Waar nieuwsgierigheid en berekening elkaar ontmoeten!","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/nl\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"nl-NL"},{"@type":"Person","@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/19b550cef1a9fbd238be112b7b7bbf64","name":"Redactioneel Team","image":{"@type":"ImageObject","inLanguage":"nl-NL","@id":"https:\/\/mathority.org\/nl\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Redactioneel Team"},"sameAs":["http:\/\/mathority.org\/nl"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/comments?post=63"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/posts\/63\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/media?parent=63"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/categories?post=63"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/nl\/wp-json\/wp\/v2\/tags?post=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}