document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190130>Question 190130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you help with another limit pleease? limit as theta tends to 0 of <BR>\n" );
document.write( "theta tan theta / 1 - cos theta </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142733 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=142733\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\theta\tan(\theta)}{1-\cos(\theta)}\"> ... Start with the given limit\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{0\tan(0)}{1-\cos(0)}\"> Evaluate the limit by plugging in 0 for each  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{0(0)}{1-1}\"> Evaluate the tangent of 0 to get 0.  Evaluate the cosine of 0 to get 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{0}{0}\"> Multiply and simplify\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\theta\tan(\theta)}{1-\cos(\theta)}=\frac{0}{0}\">\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{0}{0}\"> is an indeterminate form, this means that we must use L'Hospital's Rule to find the limit. Remember, <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule\">L'Hospital's Rule</a> states that:\r<BR>\n" );
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document.write( "If functions f and g are in some <a  rel=nofollow HREF=\"http://en.wikipedia.org/wiki/Indeterminate_form\">indeterminate form</a>, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x \to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)}\"> \r<BR>\n" );
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document.write( "So let's use L'Hospital's Rule to find the limit:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\theta\tan(\theta)}{1-\cos(\theta)}\"> ... Start with the original limit\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\tan(\theta)+\theta\sec^2(\theta)}{\sin(\theta)}\"> ... Derive the numerator and denominator separately\r<BR>\n" );
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document.write( "If you evaluate the limit for the last expression, you'll find that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\tan(\theta)+\theta\sec^2(\theta)}{\sin(\theta)}=\frac{0}{0}\">\r<BR>\n" );
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document.write( "So we must use  L'Hospital's again:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\tan(\theta)+\theta\sec^2(\theta)}{\sin(\theta)}\"> ... Start with the previous expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\sec^2(\theta)+\sec^2(\theta)+2\theta\sec(\theta)}{\cos(\theta)}\"> ... Derive the numerator and denominator separately\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{2\sec^2(\theta)+2\theta\sec(\theta)}{\cos(\theta)}\"> ... Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{2\sec^2(0)+2\cdot0\sec(0)}{\cos(0)}\"> ... Plug in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=0\"> to evaluate the limit\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{2\sec^2(0)}{\cos(0)}\"> Multiply and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =\frac{2(1)}{1}\"> Evaluate the cosine of 0 to get 1. Evaluate the secant squared of 0 to get 1/1 or just 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE =2\"> Multiply and reduce\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{\theta \to 0} \frac{\theta\tan(\theta)}{1-\cos(\theta)}=2\">\r<BR>\n" );
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document.write( "Note: you can graph to visually confirm that the limit is 2. </SPAN>\n" );
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