# How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)=x/(x^2+4)?

Jan 28, 2017

The horizontal asymptote is $y = 0$
No vertical asymptote
No oblique asymptote

#### Explanation:

The domain of $f \left(x\right)$ is ${D}_{f} \left(x\right) = \mathbb{R}$

The denominator is $> 0 \forall x \in \mathbb{R}$

There are no vertical asymptotes.

As the degree of the numerator is $<$ than the degree of the denominator, there is no oblique asymptote.

${\lim}_{x \to - \infty} f \left(x\right) = {\lim}_{x \to - \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to - \infty} \frac{1}{x} = {0}^{-}$

${\lim}_{x \to + \infty} f \left(x\right) = {\lim}_{x \to + \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to + \infty} \frac{1}{x} = {0}^{+}$

The horizontal asymptote is $y = 0$

graph{(y-x/(x^2+4))(y)=0 [-5.55, 5.55, -2.773, 2.776]}