document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204233>Question 1204233</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if limit (e ^(ln (a x)) \[Times] tan (2a/(x)))=8 , as x \[LongRightArrow] - \[Infinity]<BR>\n" );
document.write( " Find the value of a? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840358 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840358\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The natural log and base 'e' are inverses of each other.<BR>\n" );
document.write( "e^(ln(ax)) = ax\r<BR>\n" );
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document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} \to -\infty\">, then ln(ax) approaches positive infinity when a < 0.\r<BR>\n" );
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document.write( "The equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{\text{x} \to -\infty} e^{\ln(a\text{x})}\tan\left(\frac{2a}{\text{x}}\right) = 8\"><BR>\n" );
document.write( "is the same as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{\text{x} \to -\infty} a\text{x}\tan\left(\frac{2a}{\text{x}}\right) = 8\">\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax ALIGN=MIDDLE  ALT=\"ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > portion approaches positive infinity as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} \to -\infty\"> when a < 0. \r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \tan\left(\frac{2a}{\text{x}}\right)\"> portion approaches tan(0) = 0 as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} \to -\infty\">.<BR>\n" );
document.write( "This is because the 2a/x part approaches 0 as x approaches positive infinity.\r<BR>\n" );
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document.write( "This will mean<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{\text{x} \to -\infty} a\text{x}\tan\left(\frac{2a}{\text{x}}\right) = 8\"><BR>\n" );
document.write( "turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \infty*0 = 8\"><BR>\n" );
document.write( "The left hand side is one of the <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Indeterminate_form\">indeterminate forms</a> in calculus.\r<BR>\n" );
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document.write( "What you'll need to do is rewrite one of the expressions so that you have a ratio of two functions.\r<BR>\n" );
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document.write( "This is one rewrite we could do<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2Atan%28%282a%29%2Fx%29 ALIGN=MIDDLE  ALT=\"ax%2Atan%28%282a%29%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28tan%28%282a%29%2Fx%29%29%2F%281%2F%28ax%29%29 ALIGN=MIDDLE  ALT=\"%28tan%28%282a%29%2Fx%29%29%2F%281%2F%28ax%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The new equivalent expression is of the form P/Q where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+tan%28%282a%29%2Fx%29 ALIGN=MIDDLE  ALT=\"P+=+tan%28%282a%29%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q+=+1%2F%28ax%29 ALIGN=MIDDLE  ALT=\"Q+=+1%2F%28ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here, use L'Hopital's rule to apply the derivative to functions P and Q.<BR>\n" );
document.write( "Then apply the limit to see if you get another indeterminate form or not. <BR>\n" );
document.write( "If so, then apply L'Hopital's rule again.<BR>\n" );
document.write( "If not, then you'll be able to solve for 'a'.\r<BR>\n" );
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document.write( "I'll let the student take over from here.<BR>\n" );
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