# How do you find vertical, horizontal and oblique asymptotes for  f(x)= (5x-15)/(2x+4)?

Apr 23, 2016

vertical asymptote x = -2
horizontal asymptote $y = \frac{5}{2}$

#### 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 : 2x + 4 = 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

$\frac{\frac{5 x}{x} - \frac{15}{x}}{\frac{2 x}{x} + \frac{4}{x}} = \frac{5 - \frac{15}{x}}{2 + \frac{4}{x}}$

as $x \to \pm \infty , y \to \frac{5 - 0}{2 + 0}$

$\Rightarrow y = \frac{5}{2} \text{ is the asymptote }$

Oblique asymptotes occur when the degree of the numerator > degree of denominator. This is not the case here hence there are no oblique asymptotes.
graph{(5x-15)/(2x+4) [-14.24, 14.24, -7.11, 7.13]}