# How do you find vertical, horizontal and oblique asymptotes for (x^2 - 5)/( x+3)?

Apr 21, 2016

vertical asymptote x = -3
oblique asymptote y = x - 3

#### 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 + 3 = 0 $\Rightarrow x = - 3 \text{ is the asymptote }$

Horizontal asymptotes occur when the degree of the numerator ≤ degree of the denominator. This is not the case here hence there are no horizontal asymptotes.

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is the case here and to find the equation we require to use 'polynomial division'.

$\Rightarrow \frac{{x}^{2} - 5}{x + 3} = x - 3 + \frac{4}{x + 3}$

as $x \to \pm \infty , y \to x - 3$

$\Rightarrow y = x - 3 \text{ is the asymptote }$
graph{(x^2-5)/(x+3) [-46.24, 46.24, -23.12, 23.1]}