document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1030353>Question 1030353</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use Calculus. For what value of a is the following equation true?\r<BR>\n" );
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document.write( "lim ((x+a)/(x-a))^x = e<BR>\n" );
document.write( "x->inf </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #645250 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=645250\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We will use the definition of e to evaluate this: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28n-%3Einfinity%2C+%281%2Blambda%2Fn%29%5E%28n%2Flambda%29%29+=+e ALIGN=MIDDLE  ALT=\"lim%28n-%3Einfinity%2C+%281%2Blambda%2Fn%29%5E%28n%2Flambda%29%29+=+e\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now let w = x-a.<BR>\n" );
document.write( "==> x+a = w+2a, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w-%3E+infinity ALIGN=MIDDLE  ALT=\"w-%3E+infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%3Einfinity ALIGN=MIDDLE  ALT=\"x-%3Einfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28+x-%3Einfinity%2C+%28%28x%2Ba%29%2F%28x-a%29%29%5Ex+%29+=lim%28+w-%3Einfinity%2C+%28%28w%2B2a%29%2Fw%29%5E%28w%2Ba%29%29+=+lim%28+w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5E%28w%2Ba%29%29+=+lim%28+w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ew+%2A%281%2B%282a%29%2Fw%29%5Ea%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28+w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ew+%29%2Alim%28w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ea%29+=+lim%28+w-%3Einfinity%2C+%28%281%2B%282a%29%2Fw%29%5E%28w%2F%282a%29%29%29%5E%282a%29+%29%2Alim%28w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ea%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In the very last expression, the second limit <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ea%29+=+1+ ALIGN=MIDDLE  ALT=\"lim%28w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5Ea%29+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a%29%2Fw+-%3E+0 ALIGN=MIDDLE  ALT=\"%282a%29%2Fw+-%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > as w approaches infinity, and a is a fixed constant that doesn't vary with w.\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28+w-%3Einfinity%2C+%28%281%2B%282a%29%2Fw%29%5E%28w%2F%282a%29%29%29%5E%282a%29+%29+=+%28lim%28+w-%3Einfinity%2C+%281%2B%282a%29%2Fw%29%5E%28w%2F%282a%29%29%29%29%5E%282a%29++=+e%5E%282a%29+=+e ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, and hence 2a = 1, or a = 1/2. </SPAN>\n" );
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