{"id":55,"date":"2004-12-18T20:06:49","date_gmt":"2004-12-18T20:06:49","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=55"},"modified":"2004-12-18T20:06:49","modified_gmt":"2004-12-18T20:06:49","slug":"student-howlers","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=55","title":{"rendered":"Student Howlers"},"content":{"rendered":"<p>A couple of howlers seen in <a href=\"http:\/\/groups-beta.google.com\/group\/sci.math\" target=\"_blank\">sci.math<\/a> newsgroup some years ago<\/p>\n<p>1. I do like the lateral (?) thinking behind this one \ud83d\ude00 <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/741b9e33d295d2d61cdd8ef39f38cbd3.gif' title='\\displaystyle \\lim_{x \\to 0}\\frac{\\sin 7x}{5x}=\\frac{\\sin 70}{50}' alt='\\displaystyle \\lim_{x \\to 0}\\frac{\\sin 7x}{5x}=\\frac{\\sin 70}{50}' align=absmiddle><\/ul>\n<p>2. <b>Problem<\/b><\/p>\n<ul>Find <img src='\/maths\/latexrender\/pictures\/594835459cf8dc623993375e7c89aa41.gif' title='\\displaystyle \\lim_{x \\to 0}\\left(\\frac{1}{x}-\\frac{1}{\\sin x}\\right)' alt='\\displaystyle \\lim_{x \\to 0}\\left(\\frac{1}{x}-\\frac{1}{\\sin x}\\right)' align=absmiddle><\/ul>\n<p>&nbsp;&nbsp;&nbsp;<b>Answer<\/b><\/p>\n<ul>Undefined<br \/>\n<i>Proof<\/i>: <img src='\/maths\/latexrender\/pictures\/86f969cb0634ea51c49cdf7c24c93f04.gif' title='\\displaystyle \\frac{1}{x}-\\frac{1}{\\sin x}=\\frac{\\sin x - x}{x\\sin x}=\\frac{\\sin - 1}{x \\sin} \\text{ or } \\frac{\\sin - 1}{\\sin x}' alt='\\displaystyle \\frac{1}{x}-\\frac{1}{\\sin x}=\\frac{\\sin x - x}{x\\sin x}=\\frac{\\sin - 1}{x \\sin} \\text{ or } \\frac{\\sin - 1}{\\sin x}' align=absmiddle><br \/>\nTherefore, since there are two possible answers with <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle> in the denominator and you can&#8217;t get rid of it and since <img src='\/maths\/latexrender\/pictures\/9fe168ced7ae92a443518f778e54c096.gif' title='x \\to 0' alt='x \\to 0' align=absmiddle>, the answer is undefined<\/ul>\n<p>\n<a href=\"http:\/\/sixthform.info\/maths\/index.php?m=200404#23\" target=\"_blank\">Cancelling<\/a> gives an example taken from <a href=\"http:\/\/www.amazon.co.uk\/exec\/obidos\/ASIN\/1857480074\/qid%3D1082473925\/sr%3D1-1\/ref%3Dsr%5F1%5F8%5F1\/202-3869318-0022214\" target=\"_blank\">Comic Sections<\/a> by Desmond MacHale. Another one from the same book is:<\/p>\n<p>3. Solve <img src='\/maths\/latexrender\/pictures\/824593361106283b88272ac5fa91247e.gif' title='\\displaystyle \\frac{dy}{dx}=\\frac{y}{\\sin x}' alt='\\displaystyle \\frac{dy}{dx}=\\frac{y}{\\sin x}' align=absmiddle><\/p>\n<p>&nbsp;&nbsp;&nbsp;Solution:<\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/86f12bf249b60b9e62ed36c045747549.gif' title='\\displaystyle \\frac{dy}{dx}.\\frac{\\sin x}{y}=1' alt='\\displaystyle \\frac{dy}{dx}.\\frac{\\sin x}{y}=1' align=absmiddle> so <img src='\/maths\/latexrender\/pictures\/909563fbe0f2885401f0a09d49754f81.gif' title='\\displaystyle \\frac{d\\cancel{y}}{dx}.\\frac{\\sin x}{\\cancel{y}}=1' alt='\\displaystyle \\frac{d\\cancel{y}}{dx}.\\frac{\\sin x}{\\cancel{y}}=1' align=absmiddle> hence <img src='\/maths\/latexrender\/pictures\/03c35244fd6908d5dc73c0984a92c18a.gif' title='\\displaystyle \\frac{d}{dx}(\\sin x)=1' alt='\\displaystyle \\frac{d}{dx}(\\sin x)=1' align=absmiddle> thus <img src='\/maths\/latexrender\/pictures\/eec2ace9e241776c42cc3f866ed4e0d4.gif' title='\\cos x=1' alt='\\cos x=1' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/3dad28281778d5ef4b7a78c7bc7a6b09.gif' title='x = 0' alt='x = 0' align=absmiddle><\/ul>\n<p>\nDo you have any favourite howlers?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A couple of howlers seen in sci.math newsgroup some years ago 1. I do like the lateral (?) thinking behind this one \ud83d\ude00 2. Problem Find &nbsp;&nbsp;&nbsp;Answer Undefined Proof: Therefore, since there are two possible answers with in the denominator and you can&#8217;t get rid of it and since , the answer is undefined Cancelling [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-55","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/55","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=55"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/55\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=55"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=55"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=55"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}