# How do you find vertical, horizontal and oblique asymptotes for (x+6) /( 2x+1 )?

Apr 25, 2016

$x = - \frac{1}{2}$
$y = \frac{1}{2}$

#### 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 + 1 = 0 → 2x = -1 → $x = - \frac{1}{2} \text{ is the asymptote }$

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

divide terms on numerator/denominator by x

$\Rightarrow \frac{\frac{x}{x} + \frac{6}{x}}{\frac{2 x}{x} + \frac{1}{x}} = \frac{1 + \frac{6}{x}}{2 + \frac{1}{x}}$

as $x \to \pm \infty , y \to \frac{1 + 0}{2 + 0}$

$\Rightarrow y = \frac{1}{2} \text{ is the asymptote }$

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no oblique asymptotes.
graph{(x+6)/(2x+1) [-10, 10, -5, 5]}