# How do you find vertical, horizontal and oblique asymptotes for  (3x+5)/(x-6)?

May 25, 2016

#### Answer:

vertical asymptote x = 6
horizontal asymptote y = 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 : x - 6 = 0 → x = 6 is the asymptote

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

divide terms on numerator/denominator by x

$\Rightarrow \frac{\frac{3 x}{x} + \frac{5}{x}}{\frac{x}{x} - \frac{6}{x}} = \frac{3 + \frac{5}{x}}{1 - \frac{6}{x}}$

as $x \to \pm \infty , y \to \frac{3 + 0}{1 - 0}$

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

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both degree 1).Hence there are no oblique asymptotes.
graph{(3x+5)/(x-6) [-20, 20, -10, 10]}