document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=363622>Question 363622</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation: <BR>\n" );
document.write( "Evaluate the following limits or explain why they do not exist.\r<BR>\n" );
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document.write( "a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E0%2C+cotx%281-cosx%29+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E0%2C+cotx%281-cosx%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim+%28x-%3E0%2C+%28sinx%29%5Etanx+%29+ ALIGN=MIDDLE  ALT=\"+lim+%28x-%3E0%2C+%28sinx%29%5Etanx+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #259386 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=259386\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E0%2C+cotx%281-cosx%29+%29++=+lim%28x-%3E0%2C+cosx%281-cosx%29%2Fsinx+%29 ALIGN=MIDDLE  ALT=\"+lim%28x-%3E0%2C+cotx%281-cosx%29+%29++=+lim%28x-%3E0%2C+cosx%281-cosx%29%2Fsinx+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E0%2C+cosx%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+cosx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E0%2C+%281-cosx%29%2Fsinx+%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+%281-cosx%29%2Fsinx+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1*0 = 0.  The second limit was obtained by applying LHopital's rule.\r<BR>\n" );
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document.write( "b.  Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+%28sinx%29%5Etanx ALIGN=MIDDLE  ALT=\"y=+%28sinx%29%5Etanx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lny+=+tanx%2Alnsinx+=+lnsinx%2Fcotx ALIGN=MIDDLE  ALT=\"lny+=+tanx%2Alnsinx+=+lnsinx%2Fcotx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The limit as x approaches 0 has the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity%2Finfinity ALIGN=MIDDLE  ALT=\"infinity%2Finfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so apply the LH rule.  The new expression becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cosx%2Fsinx%29%2F%28-csc%5E2x%29 ALIGN=MIDDLE  ALT=\"%28cosx%2Fsinx%29%2F%28-csc%5E2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Simplifying this, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosx%2F%28-cscx%29+=+-sinxcosx ALIGN=MIDDLE  ALT=\"cosx%2F%28-cscx%29+=+-sinxcosx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  As x goes to 0, -sinx*cosx approaches 0.  Hence, lny approaches 0 as x approaches 0, and thus y approaches 1.<BR>\n" );
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