Q.Discuss the continuity of the following function at the point or on the interval indicated against it. If discontinuous, identify the type of discontinuity and state whether it is removable; if removable, redefine the function so that it becomes continuous: $f(x) = \dfrac{(x+3)(x^2-6x+8)}{x^2-x-12}$.

A removable discontinuity is the mildest kind of discontinuity a function can have at a point $x=a$: the two-sided limit $\lim_{x\to a} f(x)$ genuinely exists as a finite number, but the function fails to be continuous there only because either $f(a)$ was never defined at all, or it was assigned some other value that does not match that limit. Because the limit itself is perfectly well-behaved, the discontinuity can always be patched by simply defining (or redefining) $f(a)$ to equal $\lim_{x\to a} f(x)$; the repaired function is then continuous at $a$, and if the original function had no value there to begin with, this patched version is called its extension. The most common source of this situation is a rational expression where a factor in the numerator exactly cancels a matching factor in the denominator — the cancell …