https://file.aiquickdraw.com/mathbot/file/1726373941334in7ij3jv.jpg

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. \( f(2) = 3 \)

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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.

Find the left-hand limit, right-hand limit, overall limit, and the value of f(2) based on the given graph of the function f(x).

Learn how to determine the left-hand and right-hand limits, overall limit, and function value for a discontinuous function from a graph. This problem involves understanding the concepts of limits, continuity, and discontinuities, suitable for high school or first-year university students.

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