document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=592481>Question 592481</A>: <I><SPAN STYLE=\"word-break: break-all;\"> evaluate the limit \r<BR>\n" );
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document.write( "lim     x/arctan(5x)<BR>\n" );
document.write( "x->0\r<BR>\n" );
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document.write( "is it 1?<BR>\n" );
document.write( "x=0<BR>\n" );
document.write( "arcsin is cos(0)/sin(0)=1/0 <BR>\n" );
document.write( "so 0/1/0 is.....  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #376118 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=376118\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you substitute a zero for x into your expression you end up with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2F0 ALIGN=MIDDLE  ALT=\"0%2F0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is an example of what is called an indeterminate form. A common way to find these limits is to use what is called L'Hopital's rule. This rule says for these indeterminate forms the limit of the ratios of the derivatives will be equal to the limit of original ratio.<br><BR>\n" );
document.write( "So we find the derivatives of x and arctan(5x):<BR>\n" );
document.write( "The derivative of x is 1. and the derivative of arctan(5x) (remembering to use the chain rule) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%281+%2B+%285x%29%5E2%29%29%285%29+=+%281%2F%281+%2B+25x%5E2%29%29%2A5+=+5%2F%281%2B25x%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We can now find the limit of the original ratio by finding the limit of the ratio of these derivatives:<BR>\n" );
document.write( "lim <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%285%2F%281%2B25x%5E2%29%29 ALIGN=MIDDLE  ALT=\"1%2F%285%2F%281%2B25x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x->0<BR>\n" );
document.write( "The fraction simplifies to:<BR>\n" );
document.write( "lim <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B25x%5E2%29%2F5 ALIGN=MIDDLE  ALT=\"%281%2B25x%5E2%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x->0<BR>\n" );
document.write( "Substituting a zero for x we get a limit of 1/5.<BR>\n" );
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