Let's analyze the problem step by step:

1. Piecewise function $f(x)$ is defined as:

   \begin{cases} 4 - x^2 & \text{for } x < 4 \\ x + 4 & \text{for } x \geq 4 \end{cases}$$

2. We are tasked with finding the one-sided limit: $\lim_{x \to 4^-} g(f(x)).$

3. To solve this, we need to evaluate $f(x)$ as $x$ approaches 4 from the left ($x \to 4^-$).

   • For $x < 4$, $f(x) = 4 - x^2$.
   • Let's compute $\lim_{x \to 4^-} f(x)$: $\lim_{x \to 4^-} (4 - x^2) = 4 - (4^2) = 4 - 16 = -12.$ So, $\lim_{x \to 4^-} f(x) = -12$.

4. Now, we need to find: $\lim_{x \to 4^-} g(f(x)) = \lim_{x \to 4^-} g(-12) = g(-12).$

5. The problem asks which of the given limits matches $g(-12)$, and we can see that the correct choice corresponds to: $\lim_{x \to -12^+} g(x),$ because we are looking at the right-sided limit of $g(x)$ as $x \to -12^+$.

Thus, the correct answer is Option E: $\lim_{x \to -12^+} g(x).$

Would you like further details or have any questions?

Here are five related questions to explore further:

1. What is a piecewise function, and how do limits work for such functions?
2. How do one-sided limits differ from two-sided limits?
3. How can we compute limits involving composite functions, like $g(f(x))$?
4. What is the geometric interpretation of limits for piecewise-defined functions?
5. Why is continuity important when evaluating limits?

Tip: When working with piecewise functions, always check which piece is relevant for the point where you're taking the limit.