Analysis of the Problem

We are tasked with analyzing the behavior of $f(x)$ at $x = -4$, based on the graph provided, by determining:

1. The left-hand limit ($\lim_{x \to -4^-} f(x)$).
2. The right-hand limit ($\lim_{x \to -4^+} f(x)$).
3. The overall limit ($\lim_{x \to -4} f(x)$).
4. The continuity of $f(x)$ at $x = -4$.

Step-by-Step Solution

1. Left-hand limit ($\lim_{x \to -4^-} f(x)$)

The graph shows the behavior of $f(x)$ as $x$ approaches $-4$ from the left side:

• As $x \to -4^-$, the function values increase and approach 2.

Thus: $\lim_{x \to -4^-} f(x) = 2.$

2. Right-hand limit ($\lim_{x \to -4^+} f(x)$)

The graph shows the behavior of $f(x)$ as $x$ approaches $-4$ from the right side:

• As $x \to -4^+$, the function values decrease and approach 2.

Thus: $\lim_{x \to -4^+} f(x) = 2.$

3. Overall limit ($\lim_{x \to -4} f(x)$)

Since the left-hand limit and right-hand limit both exist and are equal: $\lim_{x \to -4^-} f(x) = \lim_{x \to -4^+} f(x) = 2,$ we conclude: $\lim_{x \to -4} f(x) = 2.$

4. Continuity of $f(x)$ at $x = -4$

For $f(x)$ to be continuous at $x = -4$, three conditions must be satisfied:

1. $f(-4)$ must exist.
2. $\lim_{x \to -4} f(x)$ must exist.
3. $f(-4)$ must equal $\lim_{x \to -4} f(x)$.

From the graph:

• The function value at $x = -4$ is given by the hollow circle at $(x, y) = (-4, 3)$, so $f(-4) = 3$.
• The overall limit $\lim_{x \to -4} f(x) = 2$.

Since $f(-4) \neq \lim_{x \to -4} f(x)$, the function is not continuous at $x = -4$.

Final Answer

1. $\lim_{x \to -4^-} f(x) = 2$
2. $\lim_{x \to -4^+} f(x) = 2$
3. $\lim_{x \to -4} f(x) = 2$
4. $f(x)$ is not continuous at $x = -4$.

Do you want me to elaborate further or analyze another point?
Here are some related questions to deepen understanding:

1. What is the difference between a limit and a function's value at a point?
2. Can a function have a limit at a point but still be discontinuous? Explain with examples.
3. How do removable and jump discontinuities differ in terms of limits and function values?
4. If $f(-4)$ were redefined to be 2, would $f(x)$ be continuous at $x = -4$?
5. How does the concept of limits help in defining derivatives?

Tip: Always compare left and right-hand limits when assessing continuity or the existence of a limit!