# How do you find the vertical, horizontal and slant asymptotes of: f(x) = (x - 3) / (x + 2)?

Aug 5, 2016

vertical asymptote x = -2
horizontal asymptote y = 1

#### Explanation:

The denominator of f(x) cannot be zero as this would make f(x) undefined.Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: x +2 = 0 $\Rightarrow x = - 2 \text{ is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide terms on numerator/denominator by x

$\frac{\frac{x}{x} - \frac{3}{x}}{\frac{x}{x} + \frac{2}{x}} = \frac{1 - \frac{3}{x}}{1 + \frac{2}{x}}$

as $x \to \pm \infty , f \left(x\right) \to \frac{1 - 0}{1 + 0}$

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

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1 ) Hence there are no slant asymptotes.
graph{(x-3)/(x+2) [-20, 20, -10, 10]}