# How to find the asymptotes of f(x) = (x^2+1)/(x^2-1) ?

Apr 11, 2016

$x = \pm 1 \mathmr{and} y = 1$.

#### Explanation:

$y = f \left(x\right) = 1 + \frac{2}{{x}^{2} - 1}$, by actual division

As $x \to \pm 1 , y \to \infty$.
As $y \to 1 , x \to \pm \infty$.
So, the lines $x = \pm 1 \mathmr{and} y = 1$ are asymptotic to the graph of y = f(x)

The graph is symmetrical about the y-axis.

It cuts y-axis at $\left(0 , - 1\right)$.

For $y > 1 , x < - 1$, for one branch, and $x > 1$, for another..

In brief, there are three branches. Two above y=1 are asymptotic to all the three asymptotes. The third through$\left(0 , - 1\right) i$ is asymptotic to y = 1 only...