# How do you find horizontal asymptotes using limits?

Aug 21, 2015

Evaluate the limits as $x$ increases without bound ($x \rightarrow \infty$) and as $x$ decreases without bound ($x \rightarrow - \infty$).

#### Explanation:

For function, $f$,

if ${\lim}_{x \rightarrow \infty} f \left(x\right) = L$ (That is, if the limit exists and is equal to the number, $L$), then the line $y = L$ is an asymptote on the right for the graph of $f$.
(If the limit fails to exist, then there is no horizontal asymptote on the right.)

if ${\lim}_{x \rightarrow - \infty} f \left(x\right) = L$ (That is, if the limit exists and is equal to the number, $L$), then the line $y = L$ is an asymptote on the left for the graph of $f$.
(If the limit fails to exist, then there is no horizontal asymptote on the left.)

For rational functions, if one of the limits at infinity exists, then the other does as well and they are equal.
Other types of functions may have an asymptot on one side, but not the other -- e.g $f \left(x\right) = {e}^{x}$
or may have different left and right horizontal asymptotes -- e.g. $f \left(x\right) = \arctan x$