Let's analyze the graph and solve each part step by step.

We are looking at the limits as $x$ approaches 3 from both directions and also at the actual value of $f(3)$. The open and closed circles in the graph will help us determine these values.

Part a: $\lim_{x \to 3^-} f(x)$

This represents the limit of $f(x)$ as $x$ approaches 3 from the left (negative direction). As we follow the graph from the left towards $x = 3$, we see that the function approaches $y = 2$.

Thus, $\lim_{x \to 3^-} f(x) = 2$

Part b: $\lim_{x \to 3^+} f(x)$

This is the limit of $f(x)$ as $x$ approaches 3 from the right (positive direction). As we follow the graph from the right towards $x = 3$, we observe that the function approaches $y = 4$.

Therefore, $\lim_{x \to 3^+} f(x) = 4$

Part c: $\lim_{x \to 3} f(x)$

The overall limit as $x$ approaches 3 exists only if the left-hand limit and right-hand limit are equal. Since $\lim_{x \to 3^-} f(x) = 2$ and $\lim_{x \to 3^+} f(x) = 4$, the left-hand and right-hand limits are not equal. Thus, the limit does not exist.

So, $\lim_{x \to 3} f(x) \text{ does not exist}.$

Part d: $f(3)$

The value of $f(x)$ at $x = 3$ is indicated by the filled or open circle at $x = 3$. There is a closed (filled) circle at $(3, 0)$, meaning that $f(3) = 0$.

Thus, $f(3) = 0.$

Final answers:

• a) $\lim_{x \to 3^-} f(x) = 2$
• b) $\lim_{x \to 3^+} f(x) = 4$
• c) $\lim_{x \to 3} f(x)$ does not exist
• d) $f(3) = 0$

Would you like further explanation on any part of this solution?

Here are 5 related questions to extend your understanding:

1. How do you define the left-hand and right-hand limits in calculus?
2. What conditions must be met for a limit to exist at a point?
3. How do you interpret open and closed circles in graphs of functions?
4. Can a function be continuous at a point if the left-hand and right-hand limits are not equal?
5. What is the significance of a function's value at a point if the limit doesn't exist?

Tip: A limit may exist at a point even if the function is not defined at that point. Keep an eye on both the limit and the actual function value when analyzing graphs.