{"id":50,"date":"2004-12-04T20:50:27","date_gmt":"2004-12-04T20:50:27","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=50"},"modified":"2004-12-04T20:50:27","modified_gmt":"2004-12-04T20:50:27","slug":"i-really-despair","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=50","title":{"rendered":"I really despair"},"content":{"rendered":"<p>I came across this answer, written over a year ago, to a question about division by zero. The answer was written by a Physics instructor who seems to be rather confused. <\/p>\n<ul><i>Division by zero is often defined as infinity.  Infinity divided by zero is infinity-squared.  For signs to be defined correctly, you must have a +0 and a -0.  This is often accomplished through the theory of limits.  The limit of x as x approaches zero from the negative side is -0.  The limit of x as x approached zero from the positive side is +0.  Higer (sic) level mathematics uses the concept of infinity quite often.<\/i><\/ul>\n<p>Let&#8217;s gloss over the first two appalling sentences. In the following sentences I suppose he is trying to refer to the fact that one can approach 0 (or any number) from different directions. If x approaches 0 from above, that is x remains positive, then you can write, for example, <img src='\/maths\/latexrender\/pictures\/dbc1e031db0aa82631fc7a7fb64783ef.gif' title='\\displaystyle \\lim_{x \\to 0^+}x=0' alt='\\displaystyle \\lim_{x \\to 0^+}x=0' align=absmiddle>. Note that the limit is written as 0 not +0 which is of course equal to 0 so there&#8217;s no point writing it! This notation allows one to show that different things can happen if x approaches 0 from above or from below and to say that <img src='\/maths\/latexrender\/pictures\/134158cc20e7c409d6c957fb7bd03174.gif' title='\\displaystyle \\lim_{x \\to 0}f(x)=l' alt='\\displaystyle \\lim_{x \\to 0}f(x)=l' align=absmiddle> if and only if <img src='\/maths\/latexrender\/pictures\/1827901340f277f073132f62fdf9a9fb.gif' title='\\displaystyle \\lim_{x \\to 0^+}f(x)=\\lim_{x \\to 0^-}{f(x)=l' alt='\\displaystyle \\lim_{x \\to 0^+}f(x)=\\lim_{x \\to 0^-}{f(x)=l' align=absmiddle> (where f is a real-valued function defined on a neighbourhood of 0).<\/p>\n<p>Personally, I like using <img src='\/maths\/latexrender\/pictures\/554818000423b5dfbcb352d9d36e75d2.gif' title='x \\searrow 0' alt='x \\searrow 0' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/41a42c0973e3c9daf00a31729310fc74.gif' title='x \\nearrow 0' alt='x \\nearrow 0' align=absmiddle> rather than <img src='\/maths\/latexrender\/pictures\/88fe90d9b0e98616849858539dd3abd1.gif' title='x \\to 0^+' alt='x \\to 0^+' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/56435cbfae82a6df0760ede19e365488.gif' title='x \\to 0^-' alt='x \\to 0^-' align=absmiddle> respectively, because they illustrate the approach from above or below.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I came across this answer, written over a year ago, to a question about division by zero. The answer was written by a Physics instructor who seems to be rather confused. Division by zero is often defined as infinity. Infinity divided by zero is infinity-squared. For signs to be defined correctly, you must have a [&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-50","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\/50","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=50"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/50\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=50"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=50"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}