{"id":6662,"date":"2011-03-25T02:15:59","date_gmt":"2011-03-25T02:15:59","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6662"},"modified":"2022-01-21T16:01:47","modified_gmt":"2022-01-21T16:01:47","slug":"determine-os-numeros-reais-a-b-e-c","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6662","title":{"rendered":"Determine os n\u00fameros reais a, b e c"},"content":{"rendered":"<p><ul id='GTTabs_ul_6662' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6662' class='GTTabs_curr'><a  id=\"6662_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6662' ><a  id=\"6662_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_6662'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Determine os n\u00fameros reais a, b e c tais que: \\[\\frac{3{{x}^{2}}-5x-7}{x-2}=ax+b+\\frac{c}{x-2}\\]<\/li>\n<li>Conjecture se o gr\u00e1fico da fun\u00e7\u00e3o racional definida por \\[f(x)=\\frac{3{{x}^{2}}-5x-7}{x-2}\\] tem uma assimptota obl\u00edqua e, no caso afirmativo, indique a sua equa\u00e7\u00e3o.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6662' onClick='GTTabs_show(1,6662)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6662'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\n<blockquote>\n<p>Determine os n\u00fameros reais a, b e c tais que: \\[\\frac{3{{x}^{2}}-5x-7}{x-2}=ax+b+\\frac{c}{x-2}\\]<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>Conjecture se o gr\u00e1fico da fun\u00e7\u00e3o racional definida por \\[f(x)=\\frac{3{{x}^{2}}-5x-7}{x-2}\\] tem uma assimptota obl\u00edqua e, no caso afirmativo, indique a sua equa\u00e7\u00e3o.<\/p>\n<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u00ad<\/p>\n<ol>\n<li>\u00a0Efetuando a divis\u00e3o do polin\u00f3mio $3{{x}^{2}}-5x-7$ por $x-2$ pela Regra de Ruffini, temos:<br \/>\n\\[\\begin{array}{*{20}{c}}<br \/>\n{}&amp;3&amp;{ &#8211; 5}&amp;{ &#8211; 7} \\\\<br \/>\n2&amp;{}&amp;6&amp;2 \\\\<br \/>\n{}&amp;3&amp;1&amp;{ &#8211; 5}<br \/>\n\\end{array}\\]<br \/>\nAssim, temos:<br \/>\n\\[\\frac{3{{x}^{2}}-5x-7}{x-2}=3x+1+\\frac{-5}{x-2}\\]<br \/>\nLogo, $a=3\\wedge b=1\\wedge c=-5$.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n\\[f(x)=3x+1+\\frac{-5}{x-2}\\Leftrightarrow \\left[ f(x)-(3x+1) \\right]=\\frac{-5}{x-2}\\]<br \/>\nQuando $x\\to -\\infty $, $\\frac{-5}{x-2}\\to {{0}^{+}}$.<\/p>\n<p>Quando $x\\to +\\infty $, $\\frac{-5}{x-2}\\to {{0}^{-}}$.<\/p>\n<p>Consequentemente, quando $x\\to -\\infty $, $\\left[ f(x)-(3x+1) \\right]\\to {{0}^{+}}$; quando $x\\to +\\infty $, $\\left[ f(x)-(3x+1) \\right]\\to {{0}^{-}}$.<\/p>\n<p>Logo, a recta de equa\u00e7\u00e3o $y=3x+1$ \u00e9 uma assimptota obl\u00edqua do gr\u00e1fico da fun\u00e7\u00e3o f.<br \/>\n\u00ad<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6663\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6663\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004.png\" data-orig-size=\"690,468\" 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=\"25-03-201 Ecra004\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004.png\" class=\"aligncenter wp-image-6663\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004.png\" alt=\"\" width=\"540\" height=\"366\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004.png 690w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004-150x101.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/25-03-201-Ecra004-400x271.png 400w\" sizes=\"auto, (max-width: 540px) 100vw, 540px\" \/><\/a><\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6662' onClick='GTTabs_show(0,6662)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine os n\u00fameros reais a, b e c tais que: \\[\\frac{3{{x}^{2}}-5x-7}{x-2}=ax+b+\\frac{c}{x-2}\\] Conjecture se o gr\u00e1fico da fun\u00e7\u00e3o racional definida por \\[f(x)=\\frac{3{{x}^{2}}-5x-7}{x-2}\\] tem uma assimptota obl\u00edqua e, no caso afirmativo, indique a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20818,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,147],"tags":[148],"series":[],"class_list":["post-6662","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-operacoes-com-funcoes","tag-operacoes-com-funcoes-2"],"views":2705,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/11V2Pag200-60_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6662","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=6662"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6662\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20818"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6662"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6662"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6662"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6662"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}