document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=984292>Question 984292</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the vertical, horizonal, and oblique asymptotes, if any , for the following  rational equation <BR>\n" );
document.write( "t(x)=x^3/x^4-81 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #605129 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=605129\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%28x%29=x%5E3%2F%28x%5E4-81%29 ALIGN=MIDDLE  ALT=\"t%28x%29=x%5E3%2F%28x%5E4-81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "recall:\r<BR>\n" );
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document.write( "if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+p%28x%29+%2F+q%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+p%28x%29+%2F+q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then:\r<BR>\n" );
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document.write( "The domain of a rational function is all real values except where the denominator, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29+=+0 ALIGN=MIDDLE  ALT=\"q%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "The roots, zeros, solutions, x-intercepts  of the rational function will be the places where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+0 ALIGN=MIDDLE  ALT=\"p%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. That is, completely ignore the denominator. Whatever makes the numerator zero will be the roots of the rational function, just like they were the roots of the polynomial function earlier.<BR>\n" );
document.write( "If you can write it in factored form, then you can tell whether it will cross or touch the x-axis at each x-intercept by whether the multiplicity on the factor is odd or even.\r<BR>\n" );
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document.write( "The equations of the vertical asymptotes can be found by finding the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29 ALIGN=MIDDLE  ALT=\"q%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters. <BR>\n" );
document.write( "The location of the horizontal asymptote is determined by looking at the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=degrees ALIGN=MIDDLE  ALT=\"degrees\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the numerator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and denominator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Cm ALIGN=MIDDLE  ALT=\"n%3Cm\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x-axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the horizontal asymptote.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m ALIGN=MIDDLE  ALT=\"n=m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=an+%2F+bm+ ALIGN=MIDDLE  ALT=\"y=an+%2F+bm+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is the horizontal asymptote. That is, the ratio of the leading coefficients.<BR>\n" );
document.write( "    If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3Em ALIGN=MIDDLE  ALT=\"n%3Em\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > horizontal asymptote. \r<BR>\n" );
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document.write( "However, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m%2B1 ALIGN=MIDDLE  ALT=\"n=m%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is an oblique or slant asymptote.\r<BR>\n" );
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document.write( " Horizontal asymptote: <BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=degrees ALIGN=MIDDLE  ALT=\"degrees\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the numerator  and denominator are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and denominator (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=4 ALIGN=MIDDLE  ALT=\"m=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3C4 ALIGN=MIDDLE  ALT=\"3%3C4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  and the x-axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the horizontal asymptote \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2F%28x%5E4-81%29-%3E0+ ALIGN=MIDDLE  ALT=\"x%5E3%2F%28x%5E4-81%29-%3E0+\"   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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x-%3E-infinity ALIGN=MIDDLE  ALT=\"+x-%3E-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " Vertical asymptote:<BR>\n" );
document.write( "find the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4-81%29=0 ALIGN=MIDDLE  ALT=\"%28x%5E4-81%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4=81 ALIGN=MIDDLE  ALT=\"x%5E4=81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4=3%5E4 ALIGN=MIDDLE  ALT=\"x%5E4=3%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%284%2Cx%29=root%284%2C3%29 ALIGN=MIDDLE  ALT=\"root%284%2Cx%29=root%284%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (two double solutions)<BR>\n" );
document.write( "so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2F%28x%5E4-81%29-%3Einfinity ALIGN=MIDDLE  ALT=\"x%5E3%2F%28x%5E4-81%29-%3Einfinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2F%28x%5E4-81%29-%3E-infinity ALIGN=MIDDLE  ALT=\"x%5E3%2F%28x%5E4-81%29-%3E-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >    as   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%3E3 ALIGN=MIDDLE  ALT=\"x-%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%3E-3 ALIGN=MIDDLE  ALT=\"x-%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=m%2B1 ALIGN=MIDDLE  ALT=\"n=m%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3C%3E4%2B1 ALIGN=MIDDLE  ALT=\"3%3C%3E4%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO ALIGN=MIDDLE  ALT=\"NO\"   DISABLED_style= \"max-width:90%; height:auto;\" > oblique or slant asymptote\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Aline%28-3%2C10%2C-3%2C-10%29%2Cline%283%2C10%2C3%2C-10%29%2C%0D%0Agraph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3%2F%28x%5E4-81%29%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );