# How do you find the vertical, horizontal or slant asymptotes for (7x+7)/(x-2)?

May 8, 2016

vertical asymptote x = 2
horizontal asymptote y = 7

#### 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 - 2 = 0 → x = 2 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

$\frac{\frac{7 x}{x} + \frac{7}{x}}{\frac{x}{x} - \frac{2}{x}} = \frac{7 + \frac{7}{x}}{1 - \frac{2}{x}}$

as $x \to \pm \infty , y \to \frac{7 + 0}{1 - 0}$

$\Rightarrow y = 7 \text{ is the asymptote }$

Slant asymptotes occur when the degree of the numerator > degree of denominator. This is not the case here (both degree 1) hence there are no slant asymptotes.
graph{(7x+7)/(x-2) [-40, 40, -20, 20]}