Question

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

Answer 1

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+8x-20=(x+10)(x-2)`
Now you can expect asymptotes at `x=-10andx=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:
`(-7x)/(x^2+8x)` as the `+5and-20` don't matter anymore

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