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.)
