Let's analyze the graph and solve the problems step by step.

Step 1: Limits as $x \to 2^-$ (Left-hand Limit)

• The limit as $x \to 2^-$ means approaching $x = 2$ from the left side.
• From the graph, as we approach 2 from the left, the function seems to approach $-2$.

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

Step 2: Limits as $x \to 2^+$ (Right-hand Limit)

• The limit as $x \to 2^+$ means approaching $x = 2$ from the right side.
• From the graph, as we approach 2 from the right, the function seems to approach $3$.

$\lim_{x \to 2^+} f(x) = 3$

Step 3: Limit as $x \to 2$

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

$\lim_{x \to 2} f(x) = \text{Does not exist}$

Step 4: Value of $f(2)$

• $f(2)$ is the value of the function at $x = 2$.
• According to the graph, there is a filled dot at $x = 2$ and $y = 3$, indicating that $f(2) = 3$.

$f(2) = 3$

Summary of Answers:

1. $\lim_{x \to 2^-} f(x) = -2$
2. $\lim_{x \to 2^+} f(x) = 3$
3. $\lim_{x \to 2} f(x) = \text{Does not exist}$
4. $f(2) = 3$

Would you like more details or further explanations on any of these steps? Here are some related questions you might explore:

1. What is the importance of left-hand and right-hand limits in calculus?
2. How do discontinuities like this affect the behavior of functions?
3. Why do the left-hand and right-hand limits need to be equal for a limit to exist at a point?
4. How can we classify different types of discontinuities in functions?
5. What are some examples of real-world phenomena that can be modeled using discontinuous functions?

Tip: Always remember, if the left-hand and right-hand limits at a point differ, the overall limit at that point does not exist.