Removing a discontinuity? How would you "remove the discontinuity" of $f$ ? In other words, how would you define $f(4)$ in order to make $f$ continuous at $x=4$?
$$f(x) = \dfrac{x^2-x-12}{x-4}$$

$f(x) = \frac{x^2-x-12}{x-4}$
$f(x) = \frac{(x-4)(x+3)}{(x-4)}$
$\lim_{x\to4}f(x) = \frac{x^2-x-12}{x-4} = \lim_{x\to4}\frac{(x-4)(x+3)}{(x-4)} = 7$
So redefine $f(x)$ as ;
$f(x) = \begin{cases}\frac{x^2-x-12}{x-4}&x\ne4\\7&x= 4\end{cases}$
