Question

What are the removable and non-removable discontinuities, if any, of `f(x)=(x-2)/(x^2 + x - 6)`?

Answer 1

`x=2` is a removable discontinuity.
`x=-3` is a non-removable discontinuity (vertical asymptote).

Explanation:

Factorizing the denominator of the function as a trinomial yields

`f(x)=(x-2)/((x+3)(x-2))`

The `(x-2)` factors hence cancel and so `x=2` is a removable discontinuity.

`x=-3` is a non-removable discontinuity and is called a vertical asymptote.
(Reason : division by zero is undefined).

The graph makes it clear :

graph{(x-2)/(x^2+x-6) [-10, 10, -5, 5]}