The horizontal line #y=b# is called a horizontal asymptote of #f(x)# if either #lim_{x to +infty}f(x)=b# or #lim_{x to -infty}f(x)=b#. In order to find horizontal asymptotes, you need to evaluate limits at infinity.
Let us find horizontal asymptotes of #f(x)={2x^2}/{1-3x^2}#.
Since
#lim_{x to +infty}{2x^2}/{1-3x^2}=lim_{x to +infty}{2x^2}/{1-3x^2}cdot{1/x^2}/{1/x^2}
=lim_{x to +infty}{2}/{1/x^2-3}=2/{0-3}=-2/3#
and
#lim_{x to -infty}{2x^2}/{1-3x^2}=lim_{x to -infty}{2x^2}/{1-3x^2}cdot{1/x^2}/{1/x^2}
=lim_{x to -infty}{2}/{1/x^2-3}=2/{0-3}=-2/3#,
#y=-2/3# is the only horizontal asymptote of #f(x)#.
(Note: In this example, there is only one horizontal asymptote since the above two limits happen to be the same, but there could be at most two horizontal asymptotes in general.)
