document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=136860>Question 136860</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5)	Find the equations for the horizontal and vertical asymptotes of the following. Type none if the function does not have an asymptote.\r<BR>\n" );
document.write( "\n" );
document.write( "a)f(x)=2x+1/x-4	 		\r<BR>\n" );
document.write( "\n" );
document.write( "Answer:<BR>\n" );
document.write( "Horizontal:<BR>\n" );
document.write( "Vertical:\r<BR>\n" );
document.write( "\n" );
document.write( "b) g(x)=3X/X^2+4	 \r<BR>\n" );
document.write( "\n" );
document.write( "Answer:<BR>\n" );
document.write( "Horizontal:<BR>\n" );
document.write( "Vertical:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "	\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #100154 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=100154\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Vertical asymptotes are defined by the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=a ALIGN=MIDDLE  ALT=\"x=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a is a value that would make the function undefined.  In these two cases, values that would make the denominator go to zero.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Horizontal asymptote.  There is either one or none for any given rational function.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the degree of the denominator polynomial is greater than the degree of the numerator polynomial, then the horiziontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the degree of the denominator polynomial is equal to the degree of the numerator polynomial, then the horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=p%2Fq ALIGN=MIDDLE  ALT=\"x=p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" > where p is the lead coefficient of the numerator and q is the lead coefficient of the denominator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the degree of the denominator is less than the degree of the numerator, then there is no horizontal asymptote.  If the degrees differ by 1, there is a straight line oblique asymptote whose equation is equal to the quotient excluding the remainder when the numerator is divided by the denominator using polynomial long division. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );