# How can i find horizontal asymptote?

Aug 30, 2014

The horizontal line $y = b$ is called a horizontal asymptote of $f \left(x\right)$ if either ${\lim}_{x \to + \infty} f \left(x\right) = b$ or ${\lim}_{x \to - \infty} f \left(x\right) = b$. In order to find horizontal asymptotes, you need to evaluate limits at infinity.

Let us find horizontal asymptotes of $f \left(x\right) = \frac{2 {x}^{2}}{1 - 3 {x}^{2}}$.
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,
$y = - \frac{2}{3}$ is the only horizontal asymptote of $f \left(x\right)$.

(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.)