N\u00e5r vi skal finde gr\u00e6ndev\u00e6rdier har vi en r\u00e6kke regneregler som vi kan bruge til at evaluerer gr\u00e6nsev\u00e6rdien.<\/p>\n\n\n\n
Herunder er der beskrevet 7 regler med eksempler.<\/p>\n\n\n\n
<\/p>\n\n\n\n
1 lov: gr\u00e6nsev\u00e6rdien for en konstant<\/strong><\/p>\n\n Hvis \\(f(x)=k\\), hvor \\(k\\) er en konstant s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} f(x)&=\\lim_{x \\to a} k\\\\ &= k\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=3\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien 1<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den f\u00f8rste regel. Vi har derfor at<\/p>\n\n $\\begin{align}\\lim_{x \\to 1} f(x) &= \\lim_{x \\to 1} 3 \\\\ &= 3\\end{align}$<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 3.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 2 lov: gr\u00e6nsev\u00e6rdien for x<\/strong><\/p>\n\n Hvis \\(f(x)=x\\), s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} f(x)&=\\lim_{x \\to a} x\\\\ &= a\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=x\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien 7<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den anden regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to 7} f(x) &= \\lim_{x \\to 7} x\\\\ &= 7\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 7.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 3 lov: gr\u00e6nsev\u00e6rdien for en funktion ganget med en konstant<\/strong><\/p>\n\n Hvis \\(g(x)=k\\cdot f(x) \\), hvor \\(k\\) er en konstant, s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} g(x)&=\\lim_{x \\to a} k\\cdot f(x)\\\\ &= k\\cdot\\lim_{x \\to a} f(x)\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=3\\cdot x\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien 2<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den tredje og anden regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to 2} f(x) &= \\lim_{x \\to 2} 3\\cdot x\\\\ &= 3\\cdot\\lim_{x\\to 2}x\\\\ &=3\\cdot 2\\\\ &=6\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 6.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 4 lov: gr\u00e6nsev\u00e6rdien for en sum eller differens<\/strong><\/p>\n\n Hvis begge af gr\u00e6nserne<\/p>\n\n \\(\\lim_{x \\to a} f(x)= L\\) og \\(\\lim_{x \\to a} g(x)=M\\)<\/p>\n\n eksistere, s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} [g(x)\\pm f(x)] &=\\lim_{x \\to a} g(x) \\pm \\lim_{x \\to a} f(x)\\\\ &=L \\pm M\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien for en sum er summen af gr\u00e6nsev\u00e6rdierne og gr\u00e6nsev\u00e6rdien for en differens er differensen af gr\u00e6nsev\u00e6rdierne.<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=x-1\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien -2<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den fjerde regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to -2} f(x) &= \\lim_{x \\to -2} x-1\\\\ &= \\lim_{x \\to -2}x -\\lim_{x \\to -2}1\\\\ &=2-1\\\\ &=1\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 1.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 5 lov: gr\u00e6nsev\u00e6rdien for et produkt<\/strong><\/p>\n\n Hvis \\(f(x)=g(x)\\cdot h(x)\\), s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} f(x)&=\\lim_{x \\to a} g(x)\\cdot h(x)\\\\ &=\\lim_{x\\to a}g(x) \\cdot \\lim_{x\\to a}h(x)\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=x^2\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien 3<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den f\u00f8rste regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to 3} f(x) &= \\lim_{x \\to 3} x^2\\\\ &=\\lim_{x \\to 3} x\\cdot x\\\\ &=\\lim_{x \\to 3} x \\cdot \\lim_{x \\to 3}\\\\ x&=3\\cdot 3\\\\ &=9 \\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 9.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 6 lov: gr\u00e6nsev\u00e6rdien for en kvotient<\/strong><\/p>\n\n Hvis \\(f(x)=\\frac{g(x)}{h(x)}\\), s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x \\to a} f(x)&=\\lim_{x \\to a} \\frac{g(x)}{h(x)}\\\\ &=\\frac{\\lim_{x\\to a}g(x)}{ \\lim_{x\\to a}h(x)}\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=\\frac{x}{x+2}\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien -4<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den f\u00f8rste regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to -4} f(x) &= \\lim_{x \\to -4} \\frac{x}{x+2}\\\\ &= \\frac{\\lim_{x\\to -4}x}{\\lim_{x\\to -4}x+2}\\\\ &=\\frac{\\lim_{x\\to -4}x}{\\lim_{x\\to -4}x-\\lim_{x\\to -4}2}\\\\ &=\\frac{-4}{-4+2}\\\\ &=\\frac{-4}{-2}\\\\ &=2\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 2.<\/p><\/div><\/div>\n<\/div><\/div>\n\n\n\n <\/p>\n\n\n\n 7 lov: gr\u00e6nsev\u00e6rdi for n\u2019te r\u00f8dder<\/strong><\/p>\n\n Hvis \\(g(x)=\\sqrt[n]{f(x)}\\), s\u00e5 vil der g\u00e6lde<\/p>\n\n \\(\\begin{align}\\lim_{x\\to a} g(x)&=\\lim_{x \\to a} \\sqrt[n]{f(x)}\\\\ &=\\sqrt[n]{\\lim_{x \\to a} f(x)}\\end{align}\\)<\/p>\n\n\n Eksempel<\/strong><\/p>\n\n Find gr\u00e6nsev\u00e6rdien for funktionen \\(f(x)=\\sqrt{x}\\) n\u00e5r \\(x\\) g\u00e5r mod v\u00e6rdien 25.<\/p>\n\n L\u00f8sning<\/p>\n\n Til at finde gr\u00e6nsev\u00e6rdien benyttes den f\u00f8rste regel. Vi har derfor at<\/p>\n\n \\(\\begin{align}\\lim_{x \\to 25} f(x) &= \\lim_{x \\to 25} \\sqrt{x}\\\\ &= \\sqrt{\\lim_{x\\to 25}x}\\\\ &=\\sqrt{25}\\\\ &=5\\end{align}\\)<\/p>\n\n Gr\u00e6nsev\u00e6rdien er derfor 5.<\/p><\/div><\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":" N\u00e5r vi skal finde gr\u00e6ndev\u00e6rdier har vi en r\u00e6kke regneregler som vi kan bruge til at evaluerer gr\u00e6nsev\u00e6rdien.<\/p>\n Der er beskrevet 7 regler med tilh\u00f8rerende eksempler.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[11,3],"tags":[15,16,74],"class_list":["post-2671","post","type-post","status-publish","format-standard","hentry","category-differentialregning","category-matematik","tag-differentialregning","tag-graensevaerdi","tag-htx"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2671","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2671"}],"version-history":[{"count":24,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2671\/revisions"}],"predecessor-version":[{"id":3718,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/2671\/revisions\/3718"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2671"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2671"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2671"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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