# How do you find the horizontal asymptote for y = (x + 1)/(x - 1)?

##### 1 Answer
Apr 1, 2018

Horizontal asymptote is $y = 1$

#### Explanation:

A vertical asymptote means when as $y \to \pm \infty$, $x$ tends to some finite number. This is simpler as we know that such a limit is brought out by the denominator, here $x - 1$. As $x - 1 \to 0$ i.e. $x \to 1$, it is apparent that $y \to = - \infty$. Hence $x = 1$ is a vertical asymptote here.

On the contrary, a horizontal asymptote means when $x \to \pm \infty$, $y$ tends to some finite number. Now as when $x \to = - \infty$, $\frac{1}{x} \to 0$, we divide numerator and denominator by $x$.

and $y = {\lim}_{x \to \infty} \frac{x + 1}{x - 1} = {\lim}_{x \to \infty} \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}$

= $1$

Hence horizontal asymptote is $y = 1$.
graph{(x+1)/(x-1) [-10, 10, -5, 5]}

Note: Observe that horizontal asymptote = is there only when degree of $x$ in numerator and denominator is same.