document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=740836>Question 740836</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For each rational function, identify any holes or horizontal or vertical asymptotes of its graph.<BR>\n" );
document.write( "1) y= (x+3)/(x+2)(x+3)<BR>\n" );
document.write( "2) y= x+5/(x-2)(x-3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #451732 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=451732\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+%28x%2B3%29%2F%28%28x%2B2%29%28x%2B3%29%29 ALIGN=MIDDLE  ALT=\"y=+%28x%2B3%29%2F%28%28x%2B2%29%28x%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+cross%28%28x%2B3%29%291%2F%28%28x%2B2%29cross%28%28x%2B3%29%29%29 ALIGN=MIDDLE  ALT=\"y=+cross%28%28x%2B3%29%291%2F%28%28x%2B2%29cross%28%28x%2B3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+1%2F%28x%2B2%29 ALIGN=MIDDLE  ALT=\"y=+1%2F%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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;\" > asymptotes will occur for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > which make the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zero ALIGN=MIDDLE  ALT=\"zero\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For this equation the denominator will be zero if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . So the vertical lines, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=-+2 ALIGN=MIDDLE  ALT=\"x+=-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be <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=+asymptote ALIGN=MIDDLE  ALT=\"+asymptote\"   DISABLED_style= \"max-width:90%; height:auto;\" > for this equation.\r<BR>\n" );
document.write( "\n" );
document.write( "For this equation, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > become very large, the fraction approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zero ALIGN=MIDDLE  ALT=\"zero\"   DISABLED_style= \"max-width:90%; height:auto;\" > in value. \r<BR>\n" );
document.write( "\n" );
document.write( "so, <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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=horizontal ALIGN=MIDDLE  ALT=\"horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptote\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28600%2C+600%2C+-10%2C+10%2C+-10%2C+20%2C++blue%28line%28-2%2C20%2C-2%2C-10%29%29%2C+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+20%2C+%28x%2B3%29%2F%28%28x%2B2%29%28x%2B3%29%29%2C0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+%28x%2B5%29%2F%28%28x-2%29%28x-3%29%29 ALIGN=MIDDLE  ALT=\"y=+%28x%2B5%29%2F%28%28x-2%29%28x-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B5%29%2F%28x%5E2-5x%2B6%29 ALIGN=MIDDLE  ALT=\"y=%28x%2B5%29%2F%28x%5E2-5x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  \r<BR>\n" );
document.write( "\n" );
document.write( "\"Holes\" would occur when the numerator and denominator have a common factor which is an expression which could be zero. <BR>\n" );
document.write( "Since this fraction has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > such <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=common ALIGN=MIDDLE  ALT=\"common\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+factors ALIGN=MIDDLE  ALT=\"+factors\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there will be no \"holes\".\r<BR>\n" );
document.write( "\n" );
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;\" > asymptotes will occur for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > which make the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zero ALIGN=MIDDLE  ALT=\"zero\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For this equation the denominator will be zero if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   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;\" >. So the vertical lines, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > will be <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;\" > for this equation.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Horizontal ALIGN=MIDDLE  ALT=\"Horizontal\"   DISABLED_style= \"max-width:90%; height:auto;\" > asymptotes will occur if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches some constant value when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > become <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=very ALIGN=MIDDLE  ALT=\"very\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+large ALIGN=MIDDLE  ALT=\"+large\"   DISABLED_style= \"max-width:90%; height:auto;\" > (positive or negative).<BR>\n" );
document.write( "For this equation, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > become very large, the denominator of the fraction becomes very large. <BR>\n" );
document.write( "This makes the fraction very, very <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=small ALIGN=MIDDLE  ALT=\"small\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In fact the fraction approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=zero ALIGN=MIDDLE  ALT=\"zero\"   DISABLED_style= \"max-width:90%; height:auto;\" > in value. <BR>\n" );
document.write( "So as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > become very large, the fraction becomes negligible and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-value ALIGN=MIDDLE  ALT=\"y-value\"   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <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;\" >.<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28600%2C+600%2C+-10%2C+10%2C+-10%2C+20%2C++blue%28line%282%2C20%2C2%2C-10%29%29%2Cblue%28line%283%2C20%2C3%2C-10%29%29%2C+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+20%2C+%28x%2B5%29%2F%28x%5E2-5x%2B6%29%2C0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );