document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1000931>Question 1000931</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please explain how to find the horizontal and vertical asymptotes of these functions using limits. If I understand correctly, to find the vertical asymptote you set the numerator to zero and solve. And to find the horizontal asymptote you compare the exponents, if they are the same, use the fraction, if the numerator is larger it DNE, if the denominator is larger then it is zero. I understand that decent enough but throwing limits in there is confusing. How do you find vertical and horizontal asymptotes by using limits on these functions?\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618315 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618315\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "If I understand correctly, to find the vertical asymptote you set the numerator to zero and solve.<BR>\n" );
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document.write( "It is incorrect.\r<BR>\n" );
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document.write( "To find the vertical asymptote you set the &nbsp;<U>denominator</U>&nbsp; to zero and solve.\r<BR>\n" );
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document.write( "===========\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+%2B+e%5E%28-x%5E2%29+ ALIGN=MIDDLE  ALT=\"5+%2B+e%5E%28-x%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "No vertical asymptotes.<BR>\n" );
document.write( "Horizontal asymptote &nbsp;y = 5, &nbsp;because &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-x%5E2%29 ALIGN=MIDDLE  ALT=\"e%5E%28-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;tends to zero &nbsp;when &nbsp;x --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===========\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3-9x%2B4%29%2F%282x%5E3%2B6x%2B7%29+ ALIGN=MIDDLE  ALT=\"%28x%5E3-9x%2B4%29%2F%282x%5E3%2B6x%2B7%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Horizontal asymptotes &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;when x ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
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document.write( "and &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > &nbsp;when x ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Vertical asymptotes are there where the denominator is zero: &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+6x+%2B+7 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+6x+%2B+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "==========\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2F%28ln%282x%2B4%29%29+ ALIGN=MIDDLE  ALT=\"%281%29%2F%28ln%282x%2B4%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "No horizontal asymptotes.\r<BR>\n" );
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document.write( "Vertical asymptote at &nbsp;x = -1.5 &nbsp;&nbsp;(where &nbsp;2x+4 = 1 &nbsp;and &nbsp;ln(2x+4) = ln(1) = 0).\r<BR>\n" );
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document.write( "The function is defined at &nbsp;x > -2 &nbsp;&nbsp;(where &nbsp;2x+4 > 0).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-3x-10%29%2F%28sqrt%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-3x-10%29%2F%28sqrt%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "No horizontal asymptotes.\r<BR>\n" );
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document.write( "Vertical asymptote &nbsp;x = -2 &nbsp;&nbsp;(where x+2 = 0).\r<BR>\n" );
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document.write( "The function is defined at &nbsp;x > -2.\r<BR>\n" );
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