# How do you find the vertical, horizontal or slant asymptotes for (x^2-4)/(x) ?

Mar 29, 2016

vertical asymptote x = 0
slant asymptote y = x

#### Explanation:

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

$\Rightarrow x = 0 \text{ is the asymptote }$

Horizontal asymptotes occur when the degree of the numerator is less than the degree of the denominator. This is not the case here hence there are no asymptotes.

slant asymptotes occur when the degree of the numerator is greater than the degree of the denominator , as in this question.

divide numerator by x

$\Rightarrow {x}^{2} / x - \frac{4}{x} = x - \frac{4}{x}$

now as x$\to \infty , \frac{4}{x} \to 0$

thus y = x is the asymptote

Here is the graph of the function.
graph{(x^2-4)/x [-10, 10, -5, 5]}