{"id":8390,"date":"2012-04-16T18:53:34","date_gmt":"2012-04-16T17:53:34","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8390"},"modified":"2021-12-30T01:10:32","modified_gmt":"2021-12-30T01:10:32","slug":"prove-que-5","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8390","title":{"rendered":"Prove que"},"content":{"rendered":"<p><ul id='GTTabs_ul_8390' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8390' class='GTTabs_curr'><a  id=\"8390_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8390' ><a  id=\"8390_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_8390'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Prove que\u00a0$$\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\operatorname{tg} x$$ n\u00e3o existe, encontrando duas sucess\u00f5es infinitamente grandes, $({u_n})$ e $({v_n})$, tais que $\\left( {\\operatorname{tg} ({u_n})} \\right)$ e $\\left( {\\operatorname{tg} ({v_n})} \\right)$ convirjam para limites diferentes.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8390' onClick='GTTabs_show(1,8390)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8390'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Consideremos as sucess\u00f5es $({u_n})$ e $({v_n})$, infinitamente grandes negativos, tais que:<\/p>\n<p>$${u_n} = \\frac{\\pi }{4} &#8211; n\\pi $$ e $${v_n} = \\frac{{3\\pi }}{4} &#8211; n\\pi $$<\/p>\n<p>Ora, $$\\mathop {\\lim }\\limits_{} \\operatorname{tg} ({u_n}) = \\mathop {\\lim }\\limits_{} \\left( {\\operatorname{tg} \\left( {\\frac{\\pi }{4} &#8211; n\\pi } \\right)} \\right) = 1$$ e $$\\mathop {\\lim }\\limits_{} \\operatorname{tg} ({v_n}) = \\mathop {\\lim }\\limits_{} \\left( {\\operatorname{tg} \\left( {\\frac{{3\\pi }}{4} &#8211; n\\pi } \\right)} \\right) =\u00a0 &#8211; 1$$<\/p>\n<p>Verifica-se que duas sucess\u00f5es de elementos pertencentes ao dom\u00ednio da fun\u00e7\u00e3o $x \\to \\operatorname{tg} x$ tendem para $ &#8211; \\infty $, enquanto as respetivas sucess\u00f5es imagens convergem para dois n\u00fameros distintos, o que contraria a defini\u00e7\u00e3o de Heine sobre o limite de uma fun\u00e7\u00e3o em $x = a$.<\/p>\n<p>Logo, n\u00e3o existe $$\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\operatorname{tg} x$$<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8390' onClick='GTTabs_show(0,8390)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Prove que\u00a0$$\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\operatorname{tg} x$$ n\u00e3o existe, encontrando duas sucess\u00f5es infinitamente grandes, $({u_n})$ e $({v_n})$, tais que $\\left( {\\operatorname{tg} ({u_n})} \\right)$ e $\\left( {\\operatorname{tg} ({v_n})} \\right)$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19664,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,300],"series":[],"class_list":["post-8390","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-funcao-tangente"],"views":2071,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat240.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8390","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=8390"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8390\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19664"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8390"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8390"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8390"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}