document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77409>Question 77409</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph of the following rational functions,give any equations for vertical,horizontal,or oblique asymptotes. Label completely! \r<BR>\n" );
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document.write( "       6+2x<BR>\n" );
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document.write( "       -4+x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55529 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=55529\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we have a 0 as our denominator, then we have a vertical asymptote, so...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Bx=0 ALIGN=MIDDLE  ALT=\"-4%2Bx=0\"   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=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our vertical asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For our horizontal asymptote, we simply evaluate x for a very large values and see where it ends up. In other words:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2B2%281000%29%29%2F%28-4%2B1000%29=2.0140562248996 ALIGN=MIDDLE  ALT=\"%286%2B2%281000%29%29%2F%28-4%2B1000%29=2.0140562248996\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let x=1000\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2B2%2810000%29%29%2F%28-4%2B10000%29=2.00140056022409 ALIGN=MIDDLE  ALT=\"%286%2B2%2810000%29%29%2F%28-4%2B10000%29=2.00140056022409\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let x=10,000\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2B2%281000000%29%29%2F%28-4%2B1000000%29=2.000014000056 ALIGN=MIDDLE  ALT=\"%286%2B2%281000000%29%29%2F%28-4%2B1000000%29=2.000014000056\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let x=1,000,000\r<BR>\n" );
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document.write( "It looks like as we let x continue on forever, y will slowly approach the value of 2. So our horizontal asymptote is y=2. It turns out that we simply divide 2x by x to get 2\r<BR>\n" );
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document.write( "And since the degrees of the numerator and the denominator are the same, we will not have any oblique asymptotes.\r<BR>\n" );
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document.write( "So here's our graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-6%2C+15%2C+-10%2C+10%2C+%286%2B2x%29%2F%28-4%2Bx%29%2C+2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-6%2C+15%2C+-10%2C+10%2C+%286%2B2x%29%2F%28-4%2Bx%29%2C+2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%286%2B2x%29%2F%28-4%2Bx%29 ALIGN=MIDDLE  ALT=\"y=%286%2B2x%29%2F%28-4%2Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The vertical line is the vertical asymptote (it is not part of the graph) and the horizontal asymptote is the green line (it is not part of the graph also, it is used to show where the asymptote is). </SPAN>\n" );
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