# How do you find the vertical, horizontal and slant asymptotes of: (3x-2) / (x+1)?

Apr 27, 2016

vertical asymptote x = -1
horizontal asymptote y = 3

#### Explanation:

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

solve : x + 1 = 0 → x = -1 , is the asymptote

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

divide terms on numerator/denominator by x

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

as $x \to \pm \infty , y \to \frac{3 - 0}{1 + 0}$

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

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.
graph{(3x-2)/(x+1) [-10, 10, -5, 5]}