# How do you find the asymptotes for f(x) = x / (3x(x-1))?

Nov 11, 2016

The vertical asymptote is $x = 1$
The horizontal asymptote is $y = 0$

#### Explanation:

You can simplify $f \left(x\right) = \frac{1}{3 \left(x - 1\right)}$
As you cannot divide by $0$, the vertical asymptote is $x = 1$

There are no slant asymptotes since the degree of the numerator $<$ the degree of the denominator

${\lim}_{x \to - \infty} f \left(x\right) = {\lim}_{x \to - \infty} \frac{1}{3 x} = {0}^{-}$

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{1}{3 x} = {0}^{+}$

So $y = 0$ is a horizontal asymptote.
graph{1/(3(x-1)) [-7.9, 7.9, -3.95, 3.95]}