document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=988782>Question 988782</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is \"lim x ->a f(x) = L\" the same as for every epsilon>0 there exists a delta >0 such that if 0<|x-a|<delta then |f(x)-L|<epsilon\r<BR>\n" );
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document.write( "Please explain<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #609263 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=609263\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{a}}\,f(x)\ =\ L\">\r<BR>\n" );
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document.write( "if and only if for all real numbers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\epsilon\,>\,0\"> there exists a real number <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\delta\,>\,0\"> such that if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 0\,<\,|x\ -\ a|\,<\,\delta\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE |f(x)\ -\ L|\,<\,\epsilon\">\r<BR>\n" );
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document.write( "This just means that I can make the difference between f(x) and L as small as I want by selecting a sufficiently small difference between x and a.\r<BR>\n" );
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document.write( "Lets say you show me the function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ 2x\ + 3\">, and I say to you that the limit as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> gets close to 1 of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\"> is 5.  You, being a properly skeptical mathematician, say \"Prove it.\"\r<BR>\n" );
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document.write( "So I say,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ |2x\ +\ 3\ -\ 5|\ <\ \epsilon\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -\epsilon\ <\ 2x\ -\ 2\ <\ \epsilon\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -\frac{\epsilon}{2}\ <\ x\ -\ 1\ <\ \frac{\epsilon}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ |x\ -\ 1|\ <\ \frac{\epsilon}{2}\">\r<BR>\n" );
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document.write( "And then I say, I have proven my case because for any positive real number you choose, no matter how small, I can make the difference between f(x) and 5 smaller than your chosen number by choosing a difference between x and a to be smaller than one-half of your number.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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