# How do you find vertical, horizontal and oblique asymptotes for (2x+3)/(3x+1) ?

May 24, 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 set the denominator equal to zero.

solve : 3x + 1 = 0 $\Rightarrow x = - \frac{1}{3} \text{ is the asymptote}$

Horizontal asymptotes occur as ${\lim}_{x \to \pm \infty} , y \to 0$

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

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

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