# How do you find the vertical, horizontal or slant asymptotes for (3x-2) /(2x+5)?

##### 1 Answer
Jun 19, 2016

vertical asymptote $x = - \frac{5}{2}$
horizontal asymptote $y = \frac{3}{2}$

#### 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 : 2x + 5 = 0 $\Rightarrow x = - \frac{5}{2}$

$\Rightarrow x = - \frac{5}{2} \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$\frac{\frac{3 x}{x} - \frac{2}{x}}{\frac{2 x}{x} + \frac{5}{x}} = \frac{3 - \frac{2}{x}}{2 + \frac{5}{x}}$

as $x \to \pm \infty , y \to \frac{3 - 0}{2 + 0}$

$\Rightarrow y = \frac{3}{2} \text{ is the asymptote}$

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