https://file.aiquickdraw.com/mathbot/file/1727228382834s25z8pc4.jpg

Let's analyze the problem step by step:

1. **Piecewise function \( f(x) \)** is defined as:
   \[
   f(x) = 
   \begin{cases} 
   4 - x^2 & \text{for } x < 4 \\
   x + 4 & \text{for } x \geq 4 
   \end{cases}
   \]

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.

Let f be the piecewise function given by f(x) = {4 - x^2 for x < 4, x + 4 for x >= 4}. Let g be an unknown function with domain (-∞,∞). Which of the following is equal to the one-sided limit lim(x -> 4^-) g(f(x))?

Learn how to evaluate the one-sided limit lim(x -> 4^-) g(f(x)) for a piecewise function f(x) and an unknown function g(x). This problem covers key calculus concepts, including piecewise functions, one-sided limits, and composite functions. Suitable for advanced high school and early college students.

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