document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=890090>Question 890090</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please evaluate : <BR>\n" );
document.write( "Lim x tends to zero, ((tanx)/x)^(1/x^2) with step by step solution. The answer is = e^(1/3) as per book.\r<BR>\n" );
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document.write( "Regard\r<BR>\n" );
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document.write( "Khoka123 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #538690 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=538690\">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\rightarrow 0}\ \left(\frac{\tan(x)}{x}\right)^{\frac{1}{x^2}}\">\r<BR>\n" );
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document.write( "Exponentiate:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow 0}\ e^{\frac{1}{x^2}\ln\left(\frac{\tan(x)}{x}\right)}\ =\ \lim_{x\rightarrow 0}\ e^{\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}\">\r<BR>\n" );
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document.write( "Using the fact that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e^x\"> is continuous at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0\"> and if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\"> is continuous around <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ g(a)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(x)\"> is continuous around <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ a\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{x\right a}\,f(g(x))\ =\ f\left( \lim_{x\rightarrow a}\,g(x)\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\rightarrow 0}\ e^{\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}}\ =\ e^{\lim_{x\right 0}\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}}\">\r<BR>\n" );
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document.write( "Use a Taylor Series expansion at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0\"> to compute <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{x\right 0}\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right 0}\,\frac{1}{3}\ +\ \frac{7x^2}{90}\ +\ O\left(x^4\right)\ =\ \frac{1}{3}\">\r<BR>\n" );
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document.write( "Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{\lim_{x\right 0}\frac{\ln\left(\frac{\tan(x)}{x}\right)}{x^2}}\ =\ e^{\frac{1}{3}}\">\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<BR>\n" );
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