Question

What are the removable and non-removable discontinuities, if any, of `f(x)=((x-3)(x+5))/((x-3)(x+3))`?

Answer 1

There is a removable discontinuity at `x=3` and a non-removable discontinuity at `x=-3`.

Explanation:

Any rational function is continuous on its domain.
Since the domain of this function is all reals except `3` and `-3`, `f` has discontinuities at `3` and `-3`.

`lim_(xrarr3)f(x) = 8/6 = 4/3` so the discontinuity at `3` is removable.

`lim_(xrarr-3)f(x)` does not exist, so the discontinuity at `-3` is non-removable.