document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117866>Question 1117866</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Without using l’Hopital’s Rule or Series find <BR>\n" );
document.write( "   Limit(tan(x))^(2x-pi) as x=pi/2  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733226 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733226\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When I try this using Perl, I get a limit of 1 when approaching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+pi+%2F+2+ ALIGN=MIDDLE  ALT=\"x+=+pi+%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the left.<BR>\n" );
document.write( "But approaching <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=+pi%2F2+ ALIGN=MIDDLE  ALT=\"+x=+pi%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the right, there seems to be a discontinuity.   Based on this data, I'd conclude the limit does not exist (the limits  Lim { f(x=a+) } and Lim{ f(x=a-) } must match to say Lim{f(x=a) } exists).<br>     <BR>\n" );
document.write( "From the left:\r<BR>\n" );
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document.write( "perl -e  'use Math::Trig; $pi=3.14159265359; $x=(31/64)*$pi; for($j=0; $j<10; $j++) { $a = tan($x)**(2*$x-$pi); $t=$x/$pi; print \"f($t * pi) = $a\n\"; $x=($x+($pi/2))/2;}' <BR>\n" );
document.write( "f(0.484375 * pi) = 0.743910236876359<BR>\n" );
document.write( "f(0.4921875 * pi) = 0.833624828514047<BR>\n" );
document.write( "f(0.49609375 * pi) = 0.897625801099151<BR>\n" );
document.write( "f(0.498046875 * pi) = 0.939405869042598<BR>\n" );
document.write( "f(0.4990234375 * pi) = 0.965115993386296<BR>\n" );
document.write( "f(0.49951171875 * pi) = 0.980316256788529<BR>\n" );
document.write( "f(0.499755859375 * pi) = 0.989057015331787<BR>\n" );
document.write( "f(0.4998779296875 * pi) = 0.993984876614831<BR>\n" );
document.write( "f(0.49993896484375 * pi) = 0.99672291920332<BR>\n" );
document.write( "f(0.499969482421875 * pi) = 0.998227432454653 <br>\r<BR>\n" );
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document.write( "From the right:\r<BR>\n" );
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document.write( "perl -e  'use Math::Trig; $pi=3.14159265359; $x=(33/64)*$pi; for($j=0; $j<10; $j++) { $a = tan($x)**(2*$x-$pi); $t=$x/$pi; print \"f($t * pi) = $a\n\"; $x=($x+($pi/2))/2;}' <BR>\n" );
document.write( "f(0.515625 * pi) = NaN    (NaN = Not a Number)<BR>\n" );
document.write( "f(0.5078125 * pi) = NaN<BR>\n" );
document.write( "f(0.50390625 * pi) = NaN<BR>\n" );
document.write( "f(0.501953125 * pi) = NaN<BR>\n" );
document.write( "f(0.5009765625 * pi) = NaN<BR>\n" );
document.write( "f(0.50048828125 * pi) = NaN<BR>\n" );
document.write( "f(0.500244140625 * pi) = NaN<BR>\n" );
document.write( "f(0.5001220703125 * pi) = NaN<BR>\n" );
document.write( "f(0.50006103515625 * pi) = NaN<BR>\n" );
document.write( "f(0.500030517578125 * pi) = NaN \r<BR>\n" );
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document.write( " </SPAN>\n" );
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