<s>[INST] 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 $x=4$?
$$f(x) = \dfrac{x^2-x-12}{x-4}$$
 [/INST] if  $ x \neq 4 $
$f(x)= \dfrac{x^2-x-12}{x-4}= \dfrac{(x-4)(x+3)}{x-4}=\dfrac{1(x+3)}{1}= x+ 3$   
so we have
f(x)=x+3
(for $ x \neq 4$)
$\lim_{x\to4}f(x) = \lim_{x\to4}(x+3) $= 7
we want to make f  continuous for this reason
the ammount of function at 4 should be equall to $$\lim_{x\to4}f(x)=7$$
so we define a function that in x=4  it equall to  $\lim_{x\to4}f(x)=7$
$$g(x)=
\begin{cases}
\dfrac{x^2-x-12}{x-4}= x+3, & x\neq 4 \\
7, & x=4
\end{cases}$$
</s>