# How do you find the vertical, horizontal or slant asymptotes for f(x)=(2x)/sqrt (9x^2-4)?

Jul 8, 2016

Vertical assymptotes at $x = \pm \frac{2}{3}$
Horizontal assymptotes at $y = \pm \frac{2}{3}$

#### Explanation:

Assymptotes appear for large $\left\mid x \right\mid$ values

For large $\left\mid x \right\mid$ values we have

$\sqrt{9 {x}^{2} - 4} \approx 3 \left\mid x \right\mid$ so for large $\left\mid x \right\mid$

$f \left(x\right) = \frac{2 x}{\sqrt{9 {x}^{2} - 4}} \approx \frac{2 x}{3 \left\mid x \right\mid} = \frac{2}{3} s i g n \left(x\right)$

Also we have vertical assymptotes at $x = \pm \frac{2}{3}$

Attached a figure