document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=294585>Question 294585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine all asymptotes(vertical,horizonalor oblique)<BR>\n" );
document.write( "f(x)= 2^2-3x+2/4x^2-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #212528 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=212528\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Vertical asymptotes occur when the denominator goes to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-1=0 ALIGN=MIDDLE  ALT=\"4x%5E2-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2=1 ALIGN=MIDDLE  ALT=\"4x%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=1%2F4 ALIGN=MIDDLE  ALT=\"x%5E2=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F2 ALIGN=MIDDLE  ALT=\"x=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The vertical asymptotes are shown below in blue. <BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".\r<BR>\n" );
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document.write( "Horizontal asymptotes occur as x goes to infinity.<BR>\n" );
document.write( "Divide numerator and denominator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282x%5E2-3x%2B2%29%2F%284x%5E2-1%29=%282-3%2Fx%2B2%2Fx%5E2%29%2F%284-1%2Fx%5E2%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282x%5E2-3x%2B2%29%2F%284x%5E2-1%29=%282-3%2Fx%2B2%2Fx%5E2%29%2F%284-1%2Fx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now as x goes to infinity,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%282-cross%283%2Fx%29%2Bcross%282%2Fx%5E2%29%29%2F%284-cross%281%2Fx%5E2%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%282-cross%283%2Fx%29%2Bcross%282%2Fx%5E2%29%29%2F%284-cross%281%2Fx%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the crossed out terms go to zero.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28+x-%3Einfinity%2C+f%28x%29+%29+%0D%0A=+2%2F4=1%2F2 ALIGN=MIDDLE  ALT=\"lim%28+x-%3Einfinity%2C+f%28x%29+%29+%0D%0A=+2%2F4=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The horizontal asymptote would be y=1/2 (shown below in green). <BR>\n" );
document.write( "There are no oblique asymptotes. <BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+-12%2C+12%2C+-3%2C+3%2C+blue%28line%28-0.5%2C-10%2C-0.5%2C10%29%29%2Cblue%28line%280.5%2C-10%2C0.5%2C10%29%29%2Cgraph%28+300%2C+300%2C+-12%2C+12%2C+-3%2C+3%2C+%282x%5E2-3x%2B2%29%2F%284x%5E2-1%29%2C1%2F2%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );