Here is the graph of $f(x)$ based on the conditions provided:

• $f(-2) = 0$.
• $\lim_{x \to -2^-} f(x) = 4$, meaning as $x$ approaches $-2$ from the left, $f(x)$ approaches 4.
• $\lim_{x \to -2^+} f(x) = 0$, meaning as $x$ approaches $-2$ from the right, $f(x)$ approaches 0.
• $\lim_{x \to -2} f(x)$ does not exist because the left and right limits are different.
• $f(-4) = 4$.
• $\lim_{x \to -4^-} f(x) = 4$, meaning as $x$ approaches $-4$ from the left, $f(x)$ remains 4.
• $\lim_{x \to -4^+} f(x) = 4$, meaning as $x$ approaches $-4$ from the right, $f(x)$ also remains 4.
• $\lim_{x \to -4} f(x) = 4$, so the limit at $-4$ exists.

Would you like more details on the graph or any of the limit concepts?

Here are 5 related questions to consider:

1. How do left-hand and right-hand limits affect the overall limit at a point?
2. What happens if the left-hand and right-hand limits do not match?
3. Can you find similar examples where a limit does not exist at certain points?
4. How do asymptotes relate to discontinuities in functions?
5. What is the role of piecewise functions in modeling real-world problems?

Tip: When dealing with limits, always check both the left-hand and right-hand behavior to determine if the overall limit exists.