# How do you find the horizontal asymptote y = (1-x^2)/(x-1)?

Sep 18, 2015

Because the numerator has degree greater than that of the denominator, there is no horizontal asymptote.

#### Explanation:

$y = \frac{1 - {x}^{2}}{x - 1}$

$= \frac{- {x}^{2} + 1}{x - 1}$

$= \frac{x \left(- x + \frac{1}{x}\right)}{x \left(1 - \frac{1}{x}\right)}$

$= \frac{- x + \frac{1}{x}}{1 - \frac{1}{x}}$

As $x$ increases without bound, $y$ decreases without bound
and
as $x$ decreases without bound, $y$ increases without bound.

The is no number $k$, and no line $y = k$ that $y$ is getting close to.

Sep 18, 2015

$y = \frac{1 - {x}^{2}}{x - 1} = - x - 1$ with exclusion $x \ne 1$

This is a line of slope $- 1$. It has no asymptotes.

#### Explanation:

$y = \frac{1 - {x}^{2}}{x - 1} = - \frac{{x}^{2} - 1}{x - 1} = - \frac{\left(x - 1\right) \left(x + 1\right)}{x - 1}$

$= - \left(x + 1\right) = - x - 1$

with exclusion $x \ne 1$

So this is a line of slope $- 1$ with an excluded point.

It has no asymptotes.