document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=927647>Question 927647</A>: <I><SPAN STYLE=\"word-break: break-all;\"> NOTE: THIS IS A CALCULUS INDETERMINATE FORM, L'HOSPITAL'S RULE PROBLEM. Any help is greatly appreciated!\r<BR>\n" );
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document.write( "Find the limit as x approaches 0 when sin 8x/ tan 9x<BR>\n" );
document.write( "I know the answer is 8/9, but I don't understand why. This is a L'Hopital rule problem, so I tried that. When I took the derivative of the top, over the derivative of the bottom I got 8 cos 8x/ 9 (sec 9x)^2. When you plug in 0 into this, you get 8/(9/0), so this would simplify to be 0/9. If a limit approaches 0/9, that means that is is a very small number being divided by a constant, so it would approach negative infinity. But that is not what it turns out to be. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #563132 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=563132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The mistake you made was thinking that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%280%29+=+0 ALIGN=MIDDLE  ALT=\"sec%280%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > when it is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%280%29+=+1 ALIGN=MIDDLE  ALT=\"sec%280%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sec%28x%29+=+1%2Fcos%28x%29 ALIGN=MIDDLE  ALT=\"sec%28x%29+=+1%2Fcos%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = \sin(8x) \"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g(x) = \tan(9x) \">\r<BR>\n" );
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document.write( "As x approaches 0, f(x) approaches 0. This is because <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to 0}f(x) = f(0) = \sin(8*0) = \sin(0) = 0 \"> <BR>\n" );
document.write( "(this limit law works because sine is a continuous function).\r<BR>\n" );
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document.write( "For similar reasons, as x approaches 0, g(x) approaches 0 (mainly because tan(x) = sin(x)/cos(x))\r<BR>\n" );
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document.write( "So as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x \to 0\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{f(x)}{g(x)} \to \frac{0}{0}\"> which is indeterminate. You are correct about that.\r<BR>\n" );
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document.write( "Because <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{f(x)}{g(x)} = \frac{0}{0}\">, which is indeterminate, we'll have to use <a  rel=nofollow HREF=\"http://mathworld.wolfram.com/LHospitalsRule.html\">L'Hopital's Rule (or L'Hospital's Rule)</a> to see if the limit exists. It may or may not.\r<BR>\n" );
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document.write( "Applying L'Hopital's Rule (or L'Hospital's Rule)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = \sin(8x) \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f'(x) = 8\cos(8x) \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g(x) = \tan(9x) \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g'(x) = 9\sec^2(9x) \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{f(x)}{g(x)} = \lim_{x\to0}\frac{f'(x)}{g'(x)} \ ... \text{LHospitals rule}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \lim_{x\to0}\frac{8\cos(8x)}{9\sec^2(9x)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \lim_{x\to0}\frac{8\cos(8x)}{9*\frac{1}{cos^2(9x)}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8\cos(8*0)}{9*\frac{1}{cos^2(9*0)}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8\cos(0)}{9*\frac{1}{cos^2(0)}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8*1}{9*\frac{1}{1^2}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8*1}{9*\frac{1}{1}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8*1}{9*1}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \lim_{x\to0}\frac{\sin(8x)}{\tan(9x)} = \frac{8}{9}\">\r<BR>\n" );
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document.write( "If you need more one-on-one help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).\r<BR>\n" );
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document.write( "Jim\r<BR>\n" );
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