Let's analyze the graph and address the limit questions one by one.

a. $\lim_{x \to 2^-} f(x)$

• As $x$ approaches 2 from the left (denoted by $x \to 2^-$), we can see from the graph that the function approaches $4$.
• Answer: 4

b. $\lim_{x \to 2^+} f(x)$

• As $x$ approaches 2 from the right (denoted by $x \to 2^+$), the graph shows that the function goes to infinity. The open circle at $x = 2$ suggests a discontinuity, and the graph increases rapidly upwards.
• Answer: $\infty$ (already checked)

c. $\lim_{x \to 2} f(x)$

• The two-sided limit requires both the left-hand and right-hand limits to be equal. Since the left-hand limit approaches 4 and the right-hand limit approaches infinity, the limit does not exist.
• Answer: DNE

d. $f(2)$

• From the graph, at $x = 2$, there is an open circle at $f(2) = 4$, and a solid point is located at $f(2) = 3$. The value of the function at this point is 3 (due to the filled dot at $(2, 3)$).
• Answer: 3

Would you like further clarification on any part of this explanation? Here are some related questions to consider:

1. What is the significance of open and closed circles on a graph in terms of limits and function values?
2. How do we distinguish between left-hand and right-hand limits on a graph?
3. Why does a discontinuity at a point affect the two-sided limit?
4. Can you explain when a limit approaches infinity and when it does not exist?
5. How does the function's behavior near discontinuities help in evaluating limits?

Tip: When evaluating two-sided limits, always ensure that both one-sided limits are equal, otherwise the two-sided limit does not exist.