How do you find the horizontal and vertical asymptotes of a function, such as (x+3)/(x^2-9)?

May 8, 2015

The vertical are $x = \pm 3$
and the horizontal is $y = 0$

The vertical asymptotes of a rational function are in the correspondence of the poles (points where the denominator is 0), because you know there the function goes to infinity on both sides (left limit and right limit) so in this case, the poles are $3$ and $- 3$, so the vertical asymptotes are

$x = 3$ and $x = - 3$
(I intend them as vertical lines: $\left(3 , y\right)$ and $\left(- 3 , y\right) \in {\mathbb{R}}^{2}$)

The horizontal asymptotes are correlated with the limit concept: if ${\lim}_{x \to + \infty} f \left(x\right) = k \in \mathbb{R}$ it means that

$\forall \epsilon \exists X > 0$ : $\forall x > X \left\mid f \left(x\right) - k \right\mid < \epsilon$

so if ${\lim}_{x \to + \infty} f \left(x\right) = k \in \mathbb{R}$ the horizontal asymptote is $y = k$
Totally symmetric is the case for $x \to - \infty$

So you know that for $x \to \infty$ (both $\pm \infty$) $f \left(x\right) \to 0$, so the horizontal asymptote is $y = 0$