{"id":11714,"date":"2014-01-28T22:58:20","date_gmt":"2014-01-28T22:58:20","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11714"},"modified":"2022-01-22T00:38:27","modified_gmt":"2022-01-22T00:38:27","slug":"equacoes-das-assintotas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11714","title":{"rendered":"Equa\u00e7\u00f5es das ass\u00edntotas"},"content":{"rendered":"<p><ul id='GTTabs_ul_11714' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11714' class='GTTabs_curr'><a  id=\"11714_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11714' ><a  id=\"11714_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_11714'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Indique, por observa\u00e7\u00e3o do gr\u00e1fico, as equa\u00e7\u00f5es das ass\u00edntotas de cada uma das seguintes fun\u00e7\u00f5es:<br \/>\n<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11715\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11715\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg\" data-orig-size=\"1313,491\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;AMMA&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1390950591&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\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3-1024x382.jpg\" class=\"aligncenter  wp-image-11715\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg\" alt=\"Gr\u00e1ficos\" width=\"788\" height=\"295\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg 1313w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3-300x112.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3-1024x382.jpg 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3-150x56.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3-400x149.jpg 400w\" sizes=\"auto, (max-width: 788px) 100vw, 788px\" \/><\/a><\/li>\n<li>Fa\u00e7a corresponder a cada um dos gr\u00e1ficos das al\u00edneas anteriores uma das seguintes fun\u00e7\u00f5es:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) = \\frac{{ &#8211; x}}{{x + 2}}}&amp;{}&amp;{g\\left( x \\right) = \\frac{{3x &#8211; 5}}{{x &#8211; 2}}}&amp;{}&amp;{h\\left( x \\right) = \\frac{{x &#8211; 2}}{x}}<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11714' onClick='GTTabs_show(1,11714)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11714'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\n<p style=\"text-align: left;\">\u00a0 \u00a0<a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/pag33-3.jpg\" alt=\"Gr\u00e1ficos\" width=\"788\" height=\"295\" \/><\/a><br \/>\n<strong>Gr\u00e1fico a)<\/strong>:<br \/>\nAss\u00edntota vertical: $x = 0$;<br \/>\nAss\u00edntota horizontal: $y = 1$.<\/p>\n<p><strong>Gr\u00e1fico b)<\/strong>:<br \/>\nAss\u00edntota vertical: $x = 2$;<br \/>\nAss\u00edntota horizontal: $y = 3$.\u00a0\u00a0 (Nota: o gr\u00e1fico est\u00e1 ligeiramente deslocado para baixo, na vertical.)<\/p>\n<p><strong>Gr\u00e1fico c)<\/strong>:<br \/>\nAss\u00edntota vertical: $x =\u00a0 &#8211; 2$;<br \/>\nAss\u00edntota horizontal: $y =\u00a0 &#8211; 1$.<\/p>\n<\/li>\n<li>\u00a0\\[\\begin{array}{*{20}{l}} \u00a0 {f\\left( x \\right) = \\frac{{ &#8211; x}}{{x + 2}}}&amp;{}&amp;{g\\left( x \\right) = \\frac{{3x &#8211; 5}}{{x &#8211; 2}}}&amp;{}&amp;{h\\left( x \\right) = \\frac{{x &#8211; 2}}{x}} \\end{array}\\]\n<p>O gr\u00e1fico a) corresponde \u00e0 fun\u00e7\u00e3o $h$: \\[h\\left( x \\right) = \\frac{{x &#8211; 2}}{x} = \\frac{x}{x} + \\frac{{ &#8211; 2}}{x} = 1 &#8211; \\frac{2}{x}\\]<br \/>\nRepare que se $x \\to {0^ &#8211; }$, ent\u00e3o $h\\left( x \\right) \\to\u00a0 + \\infty $ e se $x \\to {0^ + }$, ent\u00e3o $h\\left( x \\right) \\to\u00a0 &#8211; \\infty $.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $x = 0$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $h$.<\/p>\n<p>Repare ainda que se $x \\to\u00a0 &#8211; \\infty $, ent\u00e3o $h\\left( x \\right) \\to {1^ + }$ e se $x \\to\u00a0 + \\infty $, ent\u00e3o $f\\left( x \\right) \\to {1^ &#8211; }$.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $y = 1$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $h$ quando $x \\to\u00a0 &#8211; \\infty $ e quando $x \\to\u00a0 + \\infty $.<\/p>\n<p>O gr\u00e1fico b) corresponde \u00e0 fun\u00e7\u00e3o $g$: \\[g\\left( x \\right) = \\frac{{3x &#8211; 5}}{{x &#8211; 2}} = \\frac{{3x &#8211; 6 + 1}}{{x &#8211; 2}} = \\frac{{3\\left( {x &#8211; 2} \\right)}}{{x &#8211; 2}} + \\frac{1}{{x &#8211; 2}} = 3 + \\frac{1}{{x &#8211; 2}}\\]<br \/>\nRepare que se $x \\to {2^ &#8211; }$, ent\u00e3o $g\\left( x \\right) \\to\u00a0 &#8211; \\infty $ e se $x \\to {2^ + }$, ent\u00e3o $g\\left( x \\right) \\to\u00a0 + \\infty $.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $x = 2$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $g$.<\/p>\n<p>Repare ainda que se $x \\to\u00a0 &#8211; \\infty $, ent\u00e3o $g\\left( x \\right) \\to {3^ &#8211; }$ e se $x \\to\u00a0 + \\infty $, ent\u00e3o $f\\left( x \\right) \\to {3^ + }$.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $y = 3$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $g$ quando $x \\to\u00a0 &#8211; \\infty $ e quando $x \\to\u00a0 + \\infty $.<\/p>\n<p>O gr\u00e1fico c) corresponde \u00e0 fun\u00e7\u00e3o $f$: \\[f\\left( x \\right) = \\frac{{ &#8211; x}}{{x + 2}} = \\frac{{ &#8211; x &#8211; 2 + 2}}{{x + 2}} = \\frac{{ &#8211; x &#8211; 2}}{{x + 2}} + \\frac{2}{{x + 2}} =\u00a0 &#8211; 1 + \\frac{2}{{x + 2}}\\]<br \/>\nRepare que se $x \\to\u00a0 &#8211; {2^ &#8211; }$, ent\u00e3o $f\\left( x \\right) \\to\u00a0 &#8211; \\infty $ e se $x \\to\u00a0 &#8211; {2^ + }$, ent\u00e3o $f\\left( x \\right) \\to\u00a0 + \\infty $.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $x =\u00a0 &#8211; 2$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $f$.<\/p>\n<p>Repare ainda que se $x \\to\u00a0 &#8211; \\infty $, ent\u00e3o $f\\left( x \\right) \\to\u00a0 &#8211; {1^ &#8211; }$ e se $x \\to\u00a0 + \\infty $, ent\u00e3o $f\\left( x \\right) \\to\u00a0 &#8211; {1^ + }$.<br \/>\nLogo, a reta de equa\u00e7\u00e3o $y =\u00a0 &#8211; 1$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $f$ quando $x \\to\u00a0 &#8211; \\infty $ e quando $x \\to\u00a0 + \\infty $.<\/p>\n<p>Apresenta-se, de seguida, os gr\u00e1ficos das tr\u00eas fun\u00e7\u00f5es, representados no mesmo referencial:<\/p>\n<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11716\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11716\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7.png\" data-orig-size=\"993,673\" 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\u00e1ficos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7.png\" class=\"aligncenter  wp-image-11716\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7.png\" alt=\"Gr\u00e1ficos\" width=\"695\" height=\"471\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7.png 993w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7-300x203.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7-150x101.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-7-400x271.png 400w\" sizes=\"auto, (max-width: 695px) 100vw, 695px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11714' onClick='GTTabs_show(0,11714)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Indique, por observa\u00e7\u00e3o do gr\u00e1fico, as equa\u00e7\u00f5es das ass\u00edntotas de cada uma das seguintes fun\u00e7\u00f5es: Fa\u00e7a corresponder a cada um dos gr\u00e1ficos das al\u00edneas anteriores uma das seguintes fun\u00e7\u00f5es: \\[\\begin{array}{*{20}{l}} {f\\left(&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20860,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,125],"tags":[422,288,131],"series":[],"class_list":["post-11714","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-racionais","tag-11-o-ano","tag-assintota","tag-funcoes-racionais-2"],"views":3888,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11V2Pag33-7_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11714","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=11714"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11714\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20860"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11714"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11714"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11714"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}