Question

For the following function: g(x) = 2x + 2 / x^2 - 6x - 7 (a) Find the domain. Answer .................. (b) horizontal asymptotes ? (c) vertical asymptotes ? (d) is there discontinuity?

Answer 1

The only "forbidden" values for `x` are the ones that make the denominator equal to `0`.

`(2x+2)/(x^2-6x-7)=(2(x+1))/((x+1)(x-7)`

The domain is now limited to `x!=-1 and x!=7`
These also give the vertical asymptotes `x=-1 and x=7`
At these `x`-values there are also discontinuities , where `y` goes from `+oo` to `-oo` and back.

As for the horizontal asymptote we look at what happens when `x` gets larger and larger.
In this case we may cancel out the`(x+1)`'s to get:

`lim_(x->oo)2/(x-7)=0`

Because as `x` gets larger, the fraction will get smaller.
So the horizontal asymptote has the equation `y=0`