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

Oct 28, 2016

The vertical asymptote is $x = 6$
The horizontal asymptote is $y = 3$
There are no oblique asymptote

#### Explanation:

The function is not defined when $x = 6$ as we cannot divide by zero
So we have a vertical asymptote at $x = 6$

As the degree of the numerator is identical to the degree of the denominator, so we make a long division

$3 x + 5$$\textcolor{w h i t e}{a a a a}$∣$x - 6$
$3 x - 18$$\textcolor{w h i t e}{a a a}$∣$3$
$0 - 23$

Finally we obtain
$\frac{3 x + 5}{x - 6} = 3 + \frac{23}{x - 6}$
So $y = 3$ is a horizontal asymptote

We could get the same result by finding the limit as $x \to \pm \infty$
limit $\frac{3 x + 5}{x - 6} = 3$
$x \to \pm \infty$