Question

How to find the asymptotes of `y =(1-x^2)/( x-1)` ?

Answer 1

Notice how you can actually simplify this.

`y = (-(x^2 - 1))/(x-1)`

The numerator is a difference of two squares. `x^2 - 1 = (x+1)(x-1)`.

`= (-(x + 1)cancel((x-1)))/cancel(x-1)`

`= color(blue)(-x - 1)`

Asymptotes are generally found when the denominator of the fractional equation would be `0`. Since there are no longer points where you have to divide by `0`, there are no asymptotes.

However, since we did just cancel out `x - 1`, we do have one removable discontinuity at `(1,-2)`, since the original denominator would be undefined when `x - 1 = 0` (can't divide by `0`), and at `color(green)(x = 1)`, `color(green)(y) = -(1) - 1 = color(green)(-2)`.