document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206879>Question 206879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the horizontal and vertical asymptotes for the following functions if none type none f(x)=2x+3/x+2<BR>\n" );
document.write( "g(x)=5x/x^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 #156374 by </B><A HREF=/tutors/aboutme.mpl?userid=RAY100><B>RAY100(1637)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=RAY100><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=156374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = 3/(x+2)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Vertical  asymptote  are  zeros  of  denominator, (x+2)=0,  x= -2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Horizontal  Asymptote  is  function  of  power  on  num / den, or  n/m<BR>\n" );
document.write( "in this  case  n less  than m,  therefore  y=0  <BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "g(x) = (5x)/(x^2 -1)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Vertical  asymptote is  again  zero  of  den,,(x^2-1)=0,,,x=+/- 1<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hor  Asymptote , n less  than m, therefore  y=0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Rough  sketch  shows  f(x) starts  at  - infinity approaching y=0  from  bottom, steadily  decreases  as  it  approaches  the  asymptote  at  x=-2 at  - infinity.<BR>\n" );
document.write( "On the  positive  side  of  the  x=-2  asymptote, function  is  at  + infinity and  steadily  decreases  to  approach y=0  at + infinity <BR>\n" );
document.write( ".<BR>\n" );
document.write( ".Rough  sketch  of  g(x)  shows  three  zones  along  x  axis.<BR>\n" );
document.write( "Starting  at  x= - infinity, the  function  is  approaching y=0 from  negative  side, as function  moves  to  right it  decreases  steadily to  approach - infinity  at x=-1. On  the  other  side  of  the  x=-1 asymptote,  function  is  at - infinity,  moves  steadily upward  to  (0,0) then inverts  the  curve  and  steadily  rises  to  + infinity  at  the  x=1  asymptote. On  the  other  side  of  x=1,  function  is  at  + infinity,  dropping  steadily  to  approach  the  y=0 asymptote  at x= + infinity.<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Remember  form  of  f(x)  = a(n)x^(n) +...../b(m)x^m +....<BR>\n" );
document.write( "when n less  than m,,,,,y=o<BR>\n" );
document.write( "when n=m,,,,,y=a/b<BR>\n" );
document.write( "when n>m,,,,,,no  asymptote<BR>\n" );
document.write( "when n=(m+1),,,,slant  asymptote<BR>\n" );
document.write( "for  HORIZONTAL  Asymptotes<BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );