# How do you find the vertical, horizontal or slant asymptotes for f(x) = (2x+3)/(3x+1 )?

Apr 13, 2016

vertical asymptote $x = - \frac{1}{3}$
horizontal asymptote $y = \frac{2}{3}$

#### Explanation:

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

solve : $3 x + 1 = 0 \Rightarrow x = - \frac{1}{3}$

$\Rightarrow x = - \frac{1}{3} \text{ is the asymptote }$

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

divide terms on numerator/denominator by x

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

as $x \to \pm \infty , \frac{3}{x} \text{ and } \frac{1}{x} \to 0$

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

This is the graph of f(x).
graph{(2x+3)/(3x+1) [-10, 10, -5, 5]}