# What are the asymptotes of f(x)=-x/((x-1)(3-x)) ?

Oct 28, 2016

$f \left(x\right)$ has the following asymptotes:
Vertical at $x = 1$ and $x = 3$
Horizontal at $y = 0$ as $x \to \infty$ and at $y = 0$ as $x \to - \infty$

#### Explanation:

$f \left(x\right) = - \frac{x}{\left(x - 1\right) \left(3 - x\right)}$

We will have vertical asymptotes when the denominator is zero,
ie
$\left(x - 1\right) \left(3 - x\right) = 0 \implies x = 1 , 3$

As  x->oo => f(x) ~-x/(x(-x)),
ie  f(x) ~1/x->0^+

Similarly, As $x \to - \infty \implies f \left(x\right) \to {0}^{-}$,

$f \left(x\right)$ has the following asymptotes:
Vertical at $x = 1$ and $x = 3$
Horizontal at $y = 0$ as $x \to \infty$ and at $y = 0$ as $x \to - \infty$

graph{-x/((x-1)(3-x)) [-10, 10, -5, 5]}