Question

What are the removable and non-removable discontinuities, if any, of `f(x)=(x^2- 3x - 18)/((x^2 - 9)`?

Answer 1

Factorizing the numerator as a trinomial and the denominator as a difference of 2 squares, we may write the function as

`f(x)=((x-6)(x+3))/((x+3)(x-3))`

Hence, since division by zero is not allowed, we can conclude that `x=-3` is a removable discontinuity, also called a removable singularity, and whereas `x=3` is a non-removable discontinuity, also called a vertical asymptote.

The same result would apply were you to use long division to rewrite the function `f`.