document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=361546>Question 361546</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Please help me solve this equation: Evaluate the following limits or explain why they do not exist. <BR>\n" );
document.write( "a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E0%2C+%28xcosx-sinx%29%2Fx+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E0%2C+%28xcosx-sinx%29%2Fx+%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-%3Einfinity%2C+%28e%5Ex-2%5Ex%29%2Fx+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3Einfinity%2C+%28e%5Ex-2%5Ex%29%2Fx+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "c. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E1%2C++%28x%2F%28x-1%29+-+%281%2Flnx%29+%29+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C++%28x%2F%28x-1%29+-+%281%2Flnx%29+%29+%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 #257763 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=257763\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a.  Putting x = 0 produces the indeterminate form 0/0. But for this problem, there is NO need to use L'Hopital's rule as shown below:<BR>\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E0%2C+%28xcosx-sinx%29%2Fx%29=+lim%28x-%3E0%2C+%28xcosx%2Fx%29%29-lim%28x-%3E0%2C+sinx%2Fx%29 ALIGN=MIDDLE   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+%28cosx%29%29-lim%28x-%3E0%2C+sinx%2Fx%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E0%2C+%28cosx%29%29-lim%28x-%3E0%2C+sinx%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1 -1 = 0.  \r<BR>\n" );
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document.write( "b.   Putting x = 0 produces the indeterminate form 0/0. <BR>\n" );
document.write( "Now<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3Einfinity%2C+%28e%5Ex-2%5Ex%29%2Fx+%29++=lim%28x-%3Einfinity%2C+%28e%5Ex-%28ln2%29%2A2%5Ex%29+%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, by direct application of L'Hopital's rule.  This does NOT produce an indeterminate form, so the limit is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E0+-+%28ln2%29%2A2%5E0+=+1-ln2 ALIGN=MIDDLE  ALT=\"e%5E0+-+%28ln2%29%2A2%5E0+=+1-ln2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "c.   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E1%2C++%28x%2F%28x-1%29+-+%281%2Flnx%29+%29+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C++%28x%2F%28x-1%29+-+%281%2Flnx%29+%29+%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-%3E1%2C+%28+%28xlnx-x%2B1%29%2F%28%28x-1%29%2Alnx%29%29+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C+%28+%28xlnx-x%2B1%29%2F%28%28x-1%29%2Alnx%29%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, combining fractions.<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E1%2C+%28+lnx%2F%28lnx%2B%28x-1%29%2Fx%29%29+%29+ ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C+%28+lnx%2F%28lnx%2B%28x-1%29%2Fx%29%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, applying L-H rule. (Gives 0/0)<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E1%2C+%28+%28xlnx%29%2F%28xlnx%2Bx-1%29%29%29 ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C+%28+%28xlnx%29%2F%28xlnx%2Bx-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simplifying complex fractions, (Gives 0/0)<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lim%28x-%3E1%2C+%28+%28lnx%2B1%29%2F%28lnx%2B2%29+%29+%29 ALIGN=MIDDLE  ALT=\"+lim%28x-%3E1%2C+%28+%28lnx%2B1%29%2F%28lnx%2B2%29+%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, applying L-H rule (Not indeterminate anymore)<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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