# How do you find the vertical, horizontal or slant asymptotes for y=(13*x)/(x+34)?

Aug 27, 2017

$\text{vertical asymptote at } x = - 34$
$\text{horizontal asymptote at } y = 13$

#### Explanation:

The denominator 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.

$\text{solve "x+34=0rArrx=-34" is the asymptote}$

$\text{horizontal asymptotes occur as }$

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

$\text{divide terms on numerator/denominator by x}$

$y = \frac{\frac{13 x}{x}}{\frac{x}{x} + \frac{34}{x}} = \frac{13}{1 + \frac{34}{x}}$

as $x \to \pm \infty , y \to \frac{13}{1 + 0}$

$\Rightarrow y = 13 \text{ is the asymptote}$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{(13x)/(x+34) [-80, 80, -40, 40]}