# How to find the asymptotes of f(x) =(-7x + 5) / (x^2 + 8x -20) ?

Feb 4, 2016

There are two kinds of asymptotes. The vertical ones are when the part under the fraction bar approaches zero.

#### Explanation:

You can factor ${x}^{2} + 8 x - 20 = \left(x + 10\right) \left(x - 2\right)$
Now you can expect asymptotes at $x = - 10 \mathmr{and} x = 2$

For the horizontal asymptote we make $x$ as large as we want, both positive and negative. The function begins to look more and more like:
$\frac{- 7 x}{{x}^{2} + 8 x}$ as the $+ 5 \mathmr{and} - 20$ don't matter anymore

If we cross out the $x$'s it's more like:
$- \frac{7}{x}$ as the $+ 8$ doesn't matter as compared to the size of $x$
As $x$ gets larger, $f \left(x\right)$ gets smaller, or, in 'the language':
${\lim}_{x \to \infty} f \left(x\right) = 0 \mathmr{and} {\lim}_{x \to - \infty} f \left(x\right) = 0$
graph{(-7x+5)/(x^2+8x-20) [-32.48, 32.47, -16.23, 16.26]}