{"id":8501,"date":"2012-04-18T18:15:50","date_gmt":"2012-04-18T17:15:50","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8501"},"modified":"2021-12-30T10:02:52","modified_gmt":"2021-12-30T10:02:52","slug":"calcule-se-existir","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8501","title":{"rendered":"Calcule, se existir"},"content":{"rendered":"<p><ul id='GTTabs_ul_8501' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8501' class='GTTabs_curr'><a  id=\"8501_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8501' ><a  id=\"8501_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_8501'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Calcule, se existir:<\/p>\n<ol>\n<li>$\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 3x}}{x}$<\/li>\n<li>$\\mathop {\\lim }\\limits_{\\theta\u00a0 \\to 0} \\frac{\\theta }{{\\operatorname{sen} \\frac{\\theta }{2}}}$<\/li>\n<li>$\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 2x}}{{\\operatorname{sen} 3x}}$<\/li>\n<li>$\\mathop {\\lim }\\limits_{} \\left[ {n\\operatorname{sen} \\left( {\\frac{{2\\pi }}{n}} \\right)} \\right]$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8501' onClick='GTTabs_show(1,8501)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8501'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} x}}{x} = 1$$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,<br \/>\n$$\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 3x}}{x} = 3 \\times \\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 3x}}{{3x}} = 3 \\times \\underbrace {\\mathop {\\lim }\\limits_{y \\to 0} \\frac{{\\operatorname{sen} y}}{y}}_1 = 3$$<\/li>\n<li>Ora,<br \/>\n$$\\mathop {\\lim }\\limits_{\\theta\u00a0 \\to 0} \\frac{\\theta }{{\\operatorname{sen} \\frac{\\theta }{2}}} = \\mathop {\\lim }\\limits_{\\theta\u00a0 \\to 0} \\frac{1}{{\\frac{{\\operatorname{sen} \\frac{\\theta }{2}}}{\\theta }}} = \\frac{1}{{\\frac{1}{2} \\times \\mathop {\\lim }\\limits_{\\theta\u00a0 \\to 0} \\frac{{\\operatorname{sen} \\frac{\\theta }{2}}}{{\\frac{\\theta }{2}}}}} = \\frac{1}{{\\frac{1}{2} \\times \\underbrace {\\mathop {\\lim }\\limits_{y \\to 0} \\frac{{\\operatorname{sen} y}}{y}}_{`1}}} = 2$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 2x}}{{\\operatorname{sen} 3x}}}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to 0} \\left( {\\frac{2}{3} \\times \\frac{{\\operatorname{sen} 2x}}{{2x}} \\times \\frac{{3x}}{{\\operatorname{sen} 3x}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{2}{3} \\times \\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 2x}}{{2x}} \\times \\frac{1}{{\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 3x}}{{3x}}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{2}{3} \\times \\underbrace {\\mathop {\\lim }\\limits_{y \\to 0} \\frac{{\\operatorname{sen} y}}{y}}_1 \\times \\frac{1}{{\\underbrace {\\mathop {\\lim }\\limits_{z \\to 0} \\frac{{\\operatorname{sen} z}}{z}}_1}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{2}{3}}<br \/>\n\\end{array}$$<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{} \\left[ {n\\operatorname{sen} \\left( {\\frac{{2\\pi }}{n}} \\right)} \\right]}&amp; = &amp;{\\mathop {\\lim }\\limits_{n \\to\u00a0 + \\infty } \\left[ {n\\operatorname{sen} \\left( {\\frac{{2\\pi }}{n}} \\right)} \\right]} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{y \\to {0^ + }} \\left[ {\\frac{1}{y}\\operatorname{sen} \\left( {2\\pi y} \\right)} \\right]} \\\\<br \/>\n{}&amp; = &amp;{2\\pi\u00a0 \\times \\mathop {\\lim }\\limits_{y \\to {0^ + }} \\frac{{\\operatorname{sen} \\left( {2\\pi y} \\right)}}{{2\\pi y}}} \\\\<br \/>\n{}&amp; = &amp;{2\\pi\u00a0 \\times \\underbrace {\\mathop {\\lim }\\limits_{x \\to {0^ + }} \\frac{{\\operatorname{sen} x}}{x}}_1} \\\\<br \/>\n{}&amp; = &amp;{2\\pi }<br \/>\n\\end{array}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8501' onClick='GTTabs_show(0,8501)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Calcule, se existir: $\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 3x}}{x}$ $\\mathop {\\lim }\\limits_{\\theta\u00a0 \\to 0} \\frac{\\theta }{{\\operatorname{sen} \\frac{\\theta }{2}}}$ $\\mathop {\\lim }\\limits_{x \\to 0} \\frac{{\\operatorname{sen} 2x}}{{\\operatorname{sen} 3x}}$ $\\mathop {\\lim }\\limits_{} \\left[&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19669,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,286],"series":[],"class_list":["post-8501","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-limites"],"views":2557,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat243.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8501","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=8501"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8501\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19669"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8501"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8501"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8501"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}