We define continuity for a function "f" as the coicident lateral limit of the intermediate point.
We can traduce like this:
\[\lim_{x \rightarrow a ^{\pm}} f(x)=f(a)\]
What does this mean?
In very simple words, the sufficient and necessary condition for a function to be continous at a point as long as the lateral limits of such point have a value coincident with the image of that point.
Let's take a look at that and apply it to the problem in question, as you can see.
Since the function is divided we will study what happens on that x=-4 point.
So we will study the two following limits:
\[\lim_{x \rightarrow -4^{-}}\frac{ x+4 }{ x^2-16 }\] and
\[\lim_{x \rightarrow -4^{+}}\frac{ c }{ x+12 }\]
These two limits arise from the very definition of limits, since when \(x<-4\), the function f takes the structure of \(\frac{ x+4 }{ x^2-16 }\) and when \(x \ge -4\) the function f takes the structure \(\frac{ c }{ x+12 }\).
Now, solving those limits will give a value, and then you will want to use "c" as a variable to make them equal, therefore making the function continous.