document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116803>Question 116803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the horizontal and vertical asymptotes of the following. Type “none” if the function does not have an asymptote.\r<BR>\n" );
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document.write( "g(x)= 5x/x-1\r<BR>\n" );
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document.write( "Answer:<BR>\n" );
document.write( "Horizontal:<BR>\n" );
document.write( "Vertical:<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84927 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=84927\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The horizontal asymptote tells you, roughly, where the graph will go when x is really, really big.\r<BR>\n" );
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document.write( "When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:\r<BR>\n" );
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document.write( "y = (numerator's leading coefficient) / (denominator's leading coefficient)\r<BR>\n" );
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document.write( "This is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=always ALIGN=MIDDLE  ALT=\"always\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+true ALIGN=MIDDLE  ALT=\"+true\"   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=g%28x%29=+5x%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"g%28x%29=+5x%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >………… this function has degree-1 polynomials top and bottom; the degrees are the same in the numerator and the denominator\r<BR>\n" );
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document.write( "Since the degrees are the same, the numerator and denominator \"pull\" evenly; this graph should not drag down to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >, nor should it shoot off to infinity.<BR>\n" );
document.write( "Again, think in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=big ALIGN=MIDDLE  ALT=\"big\"   DISABLED_style= \"max-width:90%; height:auto;\" > values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=big ALIGN=MIDDLE  ALT=\"big\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you'll have, roughly, 5 times something big divided by once something big (minus a 1) so you'll roughly have <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%2Fx%29 ALIGN=MIDDLE  ALT=\"%285x%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which reduces to just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+horizontal+ ALIGN=MIDDLE  ALT=\"+horizontal+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=asymptote ALIGN=MIDDLE  ALT=\"asymptote\"   DISABLED_style= \"max-width:90%; height:auto;\" > for this is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+5 ALIGN=MIDDLE  ALT=\"+y+=+5\"   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+x ALIGN=MIDDLE  ALT=\"lim+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5x+%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"+5x+%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5 ALIGN=MIDDLE  ALT=\"+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim+x ALIGN=MIDDLE  ALT=\"lim+x\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5x+%2F+%28x+-1%29+ ALIGN=MIDDLE  ALT=\"+5x+%2F+%28x+-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > --><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5 ALIGN=MIDDLE  ALT=\"+5\"   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=Vertical+ ALIGN=MIDDLE  ALT=\"Vertical+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=asymptotes ALIGN=MIDDLE  ALT=\"asymptotes\"   DISABLED_style= \"max-width:90%; height:auto;\" > correspond to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zeroes ALIGN=MIDDLE  ALT=\"zeroes\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=denominator ALIGN=MIDDLE  ALT=\"denominator\"   DISABLED_style= \"max-width:90%; height:auto;\" > of a rational function.\r<BR>\n" );
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document.write( "So, set the denominator of the above fraction equal to zero and solve, this will tell me the values that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1+=+0 ALIGN=MIDDLE  ALT=\"x-1+=+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+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >………..so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because then I'd be dividing by zero\r<BR>\n" );
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document.write( "If you graph it, you will see how the graph avoid the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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