# How do you find all the asymptotes for function f(x)= ((3x^16)+28)/ ((15x^9)+33)?

Oct 17, 2015

$f \left(x\right)$ has a vertical asymptote $x = \sqrt[9]{- \frac{11}{5}}$

It has no horizontal or oblique asymptotes.

#### Explanation:

$f \left(x\right) = \frac{3 {x}^{16} + 28}{15 {x}^{9} + 33}$

$= \frac{1}{5} \frac{15 {x}^{16} + 140}{15 {x}^{9} + 33}$

$= \frac{1}{5} \frac{\left(15 {x}^{16} + 33 {x}^{7}\right) - 33 {x}^{7} + 140}{15 {x}^{9} + 33}$

$= {x}^{7} / 5 - \frac{33 {x}^{7} - 140}{15 \left(5 {x}^{9} + 11\right)}$

When $x = \sqrt[9]{- \frac{11}{5}}$ then denominator is zero, but the numerator is non-zero. So there is a vertical asymptote $x = \sqrt[9]{- \frac{11}{5}}$.

As $x \to \pm \infty$, $\frac{33 {x}^{7} - 140}{15 \left(5 {x}^{9} + 11\right)} \to 0$

since the $9$th power of $x$ in the denominator will dominate the $7$th power of $x$ in the numerator.

So $f \left(x\right)$ is asymptotic to ${x}^{7} / 5$ as $x \to \pm \infty$.

This is not a linear asymptote.