Question

How do you find all the asymptotes for function `f(x) = (x+1) / (x+2)`?

Answer 1

Let's start with the vertical , the value that is "forbidden" for `x`

Clearly that must be `x=-2` as this would make the denominator `=0` and that is not allowed.

The horizontal is when `x` becomes very large (either negative or positive). You will see that in that case the `+1` and `+2` will have a smaller and smaller effect on the outcome. So the whole thing tends to resemble

`x/x~~1` more and more, or: `lim_(x->+-oo) (x+1)/(x+2)=1`

So `y=1` is the horizontal asymptote:
graph{(x+1)/(x+2) [-10, 10, -5, 5]}