{"id":11717,"date":"2014-01-29T01:06:32","date_gmt":"2014-01-29T01:06:32","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11717"},"modified":"2022-01-22T00:42:51","modified_gmt":"2022-01-22T00:42:51","slug":"mais-assintotas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11717","title":{"rendered":"Mais ass\u00edntotas"},"content":{"rendered":"<p><ul id='GTTabs_ul_11717' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11717' class='GTTabs_curr'><a  id=\"11717_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11717' ><a  id=\"11717_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_11717'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Escreva as equa\u00e7\u00f5es das ass\u00edntotas dos gr\u00e1ficos das fun\u00e7\u00f5es racionais seguintes:<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) = \\frac{{ &#8211; 2}}{{x &#8211; 2}}}&amp;{}&amp;{g\\left( x \\right) = \\frac{{x &#8211; 7}}{{x + 2}}}&amp;{}&amp;{h\\left( x \\right) = \\frac{{3x &#8211; 3}}{{2x + 4}}}<br \/>\n\\end{array}\\]<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11717' onClick='GTTabs_show(1,11717)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11717'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>\\[\\begin{array}{*{20}{l}}<br \/>\n{f\\left( x \\right) = \\frac{{ &#8211; 2}}{{x &#8211; 2}}}&amp;{}&amp;{g\\left( x \\right) = \\frac{{x &#8211; 7}}{{x + 2}}}&amp;{}&amp;{h\\left( x \\right) = \\frac{{3x &#8211; 3}}{{2x + 4}}}<br \/>\n\\end{array}\\]<\/p>\n<\/p>\n<hr \/>\n<h4>Fun\u00e7\u00e3o $f$<\/h4>\n<p>\u00a0Ora, \\[f\\left( x \\right) = \\frac{{ &#8211; 2}}{{x &#8211; 2}} = 0 &#8211; \\frac{2}{{x &#8211; 2}}\\]<\/p>\n<p><span style=\"text-decoration: underline;\">Ass\u00edntota vertical<\/span>:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to {2^ &#8211; }} f\\left( x \\right) =\u00a0 + \\infty }&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to {2^ + }} f\\left( x \\right) =\u00a0 &#8211; \\infty }<br \/>\n\\end{array}\\]<\/p>\n<p>Logo,\u00a0a reta de equa\u00e7\u00e3o $x = 2$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $f$.<\/p>\n<p>\u00a0<span style=\"text-decoration: underline;\">Ass\u00edntota horizontal<\/span>:<\/p>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } f\\left( x \\right) = {0^ + }}&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f\\left( x \\right) = {0^ &#8211; }}<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, a reta de equa\u00e7\u00e3o $y = 0$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $f$, quando ${x \\to\u00a0 &#8211; \\infty }$ e quando ${x \\to\u00a0 + \\infty }$.<\/p>\n<\/p>\n<hr \/>\n<h4>Fun\u00e7\u00e3o $g$<\/h4>\n<p>\u00a0Ora, \\[g\\left( x \\right) = \\frac{{x &#8211; 7}}{{x + 2}} = \\frac{{x + 2 &#8211; 9}}{{x + 2}} = \\frac{{x + 2}}{{x + 2}} + \\frac{{ &#8211; 9}}{{x + 2}} = 1 &#8211; \\frac{9}{{x + 2}}\\]<\/p>\n<p><span style=\"text-decoration: underline;\">Ass\u00edntota vertical<\/span>:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; {2^ &#8211; }} g\\left( x \\right) =\u00a0 + \\infty }&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; {2^ + }} g\\left( x \\right) =\u00a0 &#8211; \\infty }<br \/>\n\\end{array}\\]<\/p>\n<p>Logo,\u00a0a reta de equa\u00e7\u00e3o $x =\u00a0 &#8211; 2$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $g$.<\/p>\n<p>\u00a0<span style=\"text-decoration: underline;\">Ass\u00edntota horizontal<\/span>:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } g\\left( x \\right) = {1^ + }}&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } g\\left( x \\right) = {1^ &#8211; }}<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, a reta de equa\u00e7\u00e3o $y = 1$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $g$, quando ${x \\to\u00a0 &#8211; \\infty }$ e quando ${x \\to\u00a0 + \\infty }$.<\/p>\n<\/p>\n<hr \/>\n<h4>Fun\u00e7\u00e3o $h$<\/h4>\n<p>\u00a0Ora, \\[h\\left( x \\right) = \\frac{{3x &#8211; 3}}{{2x + 4}} = \\frac{{3\\left( {x &#8211; 1} \\right)}}{{2\\left( {x + 2} \\right)}} = \\frac{{3\\left( {x + 2 &#8211; 3} \\right)}}{{2\\left( {x + 2} \\right)}} = \\frac{3}{2}\\left( {\\frac{{x + 2}}{{x + 2}} + \\frac{{ &#8211; 3}}{{x + 2}}} \\right) = \\frac{3}{2}\\left( {1 &#8211; \\frac{3}{{x + 2}}} \\right) = \\frac{3}{2} &#8211; \\frac{{\\frac{9}{2}}}{{x + 2}}\\]<\/p>\n<p><span style=\"text-decoration: underline;\">Ass\u00edntota vertical<\/span>:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; {2^ &#8211; }} h\\left( x \\right) =\u00a0 + \\infty }&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; {2^ + }} h\\left( x \\right) =\u00a0 &#8211; \\infty }<br \/>\n\\end{array}\\]<\/p>\n<p>Logo,\u00a0a reta de equa\u00e7\u00e3o $x =\u00a0 &#8211; 2$ \u00e9 ass\u00edntota vertical bilateral do gr\u00e1fico de $h$.<\/p>\n<p>\u00a0<span style=\"text-decoration: underline;\">Ass\u00edntota horizontal<\/span>:<\/p>\n<p>\u00a0\\[\\begin{array}{*{20}{c}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } h\\left( x \\right) = {{\\tfrac{3}{2}}^ + }}&amp;{\\text{e}}&amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } h\\left( x \\right) = {{\\tfrac{3}{2}}^ &#8211; }}<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, a reta de equa\u00e7\u00e3o $y = \\frac{3}{2}$ \u00e9 ass\u00edntota horizontal do gr\u00e1fico de $h$, quando ${x \\to\u00a0 &#8211; \\infty }$ e quando ${x \\to\u00a0 + \\infty }$.<\/p>\n<\/p>\n<p>Apresenta-se de seguida os gr\u00e1ficos das tr\u00eas fun\u00e7\u00f5es:<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"11718\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=11718\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8.png\" data-orig-size=\"1401,826\" 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-33-8-1024x603.png\" class=\"aligncenter wp-image-11718\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8.png\" alt=\"Gr\u00e1ficos\" width=\"960\" height=\"566\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8.png 1401w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8-300x176.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8-1024x603.png 1024w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8-150x88.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11-pag-33-8-400x235.png 400w\" sizes=\"auto, (max-width: 960px) 100vw, 960px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11717' onClick='GTTabs_show(0,11717)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Escreva as equa\u00e7\u00f5es das ass\u00edntotas dos gr\u00e1ficos das fun\u00e7\u00f5es racionais seguintes: \\[\\begin{array}{*{20}{l}} {f\\left( x \\right) = \\frac{{ &#8211; 2}}{{x &#8211; 2}}}&amp;{}&amp;{g\\left( x \\right) = \\frac{{x &#8211; 7}}{{x + 2}}}&amp;{}&amp;{h\\left( x \\right)&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20861,"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-11717","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":2992,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/01\/11V2Pag33-8_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11717","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=11717"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11717\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20861"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11717"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11717"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11717"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11717"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}