Let's address the problem step by step using the description of the graph of the function $f(x)$.

(a) $\lim_{x \to 4} f(x)$

The limit $\lim_{x \to 4} f(x)$ represents the value the function $f(x)$ approaches as $x$ gets closer and closer to 4 from both the left and right sides.

According to the description, the graph of the function passes through the point $(4, -8)$, and no jumps or discontinuities are mentioned around $x = 4$. This implies that the function is continuous at $x = 4$. Therefore, the limit is equal to the function value at this point.

$\lim_{x \to 4} f(x) = -8$

(b) $\lim_{x \to -4} f(x)$

Similarly, the limit $\lim_{x \to -4} f(x)$ represents the value the function approaches as $x$ gets closer to $-4$. From the description, the graph passes through the point $(-4, -4)$, and there are no jumps or discontinuities around $x = -4$, suggesting the function is also continuous at this point.

$\lim_{x \to -4} f(x) = -4$

Summary:

• $\lim_{x \to 4} f(x) = -8$
• $\lim_{x \to -4} f(x) = -4$

Let me know if you'd like more details or have any questions! Here are five related questions that might help expand your understanding:

1. What does it mean for a function to be continuous at a point?
2. How can limits help in identifying discontinuities in a graph?
3. What is the formal definition of a limit?
4. How do left-hand and right-hand limits affect the existence of a limit?
5. What are the common types of discontinuities in a function?

Tip: When analyzing a limit from a graph, always check the behavior of the function from both sides of the point of interest. If the function approaches the same value from both directions, the limit exists and equals that value!