Question

How do you find the horizontal asymptote for `y = (x + 1)/(x - 1)`?

Answer 1

Horizontal asymptote is `y=1`

Explanation:

A vertical asymptote means when as `y->+-oo`, `x` tends to some finite number. This is simpler as we know that such a limit is brought out by the denominator, here `x-1`. As `x-1->0` i.e. `x->1`, it is apparent that `y->=-oo`. Hence `x=1` is a vertical asymptote here.

On the contrary, a horizontal asymptote means when `x->+-oo`, `y` tends to some finite number. Now as when `x->=-oo`, `1/x->0`, we divide numerator and denominator by `x`.

and `y=lim_(x->oo)(x+1)/(x-1)=lim_(x->oo)(1+1/x)/(1-1/x)`

= `1`

Hence horizontal asymptote is `y=1`.
graph{(x+1)/(x-1) [-10, 10, -5, 5]}

Note: Observe that horizontal asymptote = is there only when degree of `x` in numerator and denominator is same.