# How do you find the vertical, horizontal or slant asymptotes for f(x)=x/(x-1)^2?

Jan 26, 2017

The vertical asymptote is $x = 1$
The horizontal asymptote is $y = 0$
No slant asymptote

#### Explanation:

As you cannot divide by $0$, $x \ne 1$

The vertical asymptote is $x = 1$

As the degree of the numerator is $<$ than the degree of the denominator, there is no slant asymptote.

${\lim}_{x \to - \infty} f \left(x\right) = {\lim}_{x \to - \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to - \infty} \frac{1}{x} = {0}^{-}$

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to + \infty} \frac{1}{x} = {0}^{+}$

The horizontal asymptote is $y = 0$

graph{(y-x/(x-1)^2)(y)(y-100x+100)=0 [-4.38, 4.39, -2.19, 2.193]}