# How do you find the vertical, horizontal or slant asymptotes for f(x) = (3x)/(x-8)?

Apr 3, 2016

vertical asymptote x = 8
horizontal asymptote y = 3

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve : x - 8 = 0 → x = 8 is the asymptote

Horizontal asymptotes occur as ${\lim}_{x \to \pm \infty} f \left(x\right) \to 0$

divide all terms on numerator/denominator by x

$\frac{\frac{3 x}{x}}{\frac{x}{x} - \frac{8}{x}} = \frac{3}{1 - \frac{8}{x}}$

As x$\to \pm \infty , \frac{8}{x} \to 0$

$\Rightarrow y = \frac{3}{1} = 3 \text{ is the asymptote }$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.

Here is the graph of f(x)
graph{(3x)/(x-8) [-10, 10, -5, 5]}