# How do you find the asymptotes for f(x) = (3x + 5) / (2x - 3)?

Feb 11, 2016

Vertical asymptote at $x = \frac{3}{2}$.

Horizontal asymptote at $y = \frac{3}{2}$.

#### Explanation:

Vertical asymptotes occur when the denominator is zero,
ie when $2 x - 3 = 0 \implies x = \frac{3}{2}$.

Horizontal asymptotes occur at
${\lim}_{x \to \pm \infty} f \left(x\right) = {\lim}_{x \to \pm \infty} \frac{3 x + 5}{2 x - 3} = \frac{3}{2}$

The graph of the rational function verifies this :

graph{(3x+5)/(2x-3) [-13.3, 18.73, -6.8, 9.22]}