# How do you find the vertical, horizontal and slant asymptotes of: y= (3x+5)/(x-6)?

Dec 29, 2016

vertical asymptote at x =6
horizontal asymptote at y = 3

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : $x - 6 = 0 \Rightarrow x = 6 \text{ is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , y \to c \text{ ( a constant)}$

divide terms on numerator/denominator by x

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

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

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

Slant 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 slant asymptotes.
graph{(3x+5)/(x-6) [-20, 20, -10, 10]}