# How do you find the slant asymptote of  ( x^4 + 1 ) / ( x^2 + 2 )?

##### 1 Answer
Jan 15, 2016

This rational function is asymptotic to a parabola, not a line.

It has no slant asymptote.

#### Explanation:

The degree of the numerator is $4$ and the degree of the numerator is $2$.

As a result this rational function is asymptotic to a parabola, not a line.

More explicitly:

$f \left(x\right) = \frac{{x}^{4} + 1}{{x}^{2} + 2}$

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

$= \frac{{x}^{2} \left({x}^{2} + 2\right) - 2 \left({x}^{2} + 2\right) + 5}{{x}^{2} + 2}$

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

So as $x \to \pm \infty$ we find $\left(f \left(x\right) - \left({x}^{2} - 2\right)\right) \to 0$

That is $f \left(x\right)$ is asymptotic to ${x}^{2} - 2$