{"id":7406,"date":"2012-03-08T01:46:21","date_gmt":"2012-03-08T01:46:21","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7406"},"modified":"2021-12-28T01:18:17","modified_gmt":"2021-12-28T01:18:17","slug":"mostre-que-a-reta-de-equacao-y-2x-1-e-assintota-do-grafico-da-funcao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7406","title":{"rendered":"Mostre que a reta de equa\u00e7\u00e3o $y = 2x &#8211; 1$ \u00e9 ass\u00edntota do gr\u00e1fico da fun\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_7406' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7406' class='GTTabs_curr'><a  id=\"7406_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7406' ><a  id=\"7406_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7406'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Mostre que a reta de equa\u00e7\u00e3o $y = 2x &#8211; 1$ \u00e9 ass\u00edntota do gr\u00e1fico da fun\u00e7\u00e3o $$f:x \\to \\frac{{2{x^3} &#8211; {x^2} &#8211; x + 1}}{{{x^2} &#8211; 1}}$$<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7406' onClick='GTTabs_show(1,7406)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7406'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Ora, \\[\\begin{array}{*{20}{l}}{\\mathop {\\lim }\\limits_{x \\to \u00a0+ \\infty } \\left[ {f(x) &#8211; \\left( {2x &#8211; 1} \\right)} \\right]}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to \u00a0+ \\infty } \\left[ {\\frac{{2{x^3} &#8211; {x^2} &#8211; x + 1}}{{{x^2} &#8211; 1}} &#8211; \\left( {2x &#8211; 1} \\right)} \\right]}\\\\{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to \u00a0+ \\infty } \\frac{{2{x^3} &#8211; {x^2} &#8211; x + 1 &#8211; 2{x^3} + {x^2} + 2x &#8211; 1}}{{{x^2} &#8211; 1}}}\\\\{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to \u00a0+ \\infty } \\frac{x}{{{x^2} &#8211; 1}}}\\\\{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to \u00a0+ \\infty } \\frac{{\\frac{1}{x}}}{{1 &#8211; \\frac{1}{{{x^2}}}}}}\\\\{}&amp; = &amp;0\\end{array}\\]<\/p>\n<p>Como $$\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\left[ {f(x) &#8211; \\left( {2x &#8211; 1} \\right)} \\right] = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left[ {f(x) &#8211; \\left( {2x &#8211; 1} \\right)} \\right] = 0$$ ent\u00e3o a reta de equa\u00e7\u00e3o $y = 2x &#8211; 1$ \u00e9 ass\u00edntota do gr\u00e1fico de $f$, quando ${x \\to\u00a0 &#8211; \\infty }$ e quando ${x \\to\u00a0 + \\infty }$.<\/p>\n<p>\u00a0<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7407\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7407\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b.png\" data-orig-size=\"648,402\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b.png\" class=\"aligncenter wp-image-7407 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b.png\" alt=\"\" width=\"648\" height=\"402\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b.png 648w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b-300x186.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b-150x93.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12-pag207-23b-400x248.png 400w\" sizes=\"auto, (max-width: 648px) 100vw, 648px\" \/><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7406' onClick='GTTabs_show(0,7406)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Mostre que a reta de equa\u00e7\u00e3o $y = 2x &#8211; 1$ \u00e9 ass\u00edntota do gr\u00e1fico da fun\u00e7\u00e3o $$f:x \\to \\frac{{2{x^3} &#8211; {x^2} &#8211; x + 1}}{{{x^2} &#8211; 1}}$$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19497,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,285],"tags":[427,288,286],"series":[],"class_list":["post-7406","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-teoria-de-limites","tag-12-o-ano","tag-assintota","tag-limites"],"views":3658,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/Grafico_2P207-Ex23.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7406","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7406"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7406\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19497"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7406"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7406"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7406"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}