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answer all of the questions below limx->-2-f(x)    limx->-2+f(x)    f(-2)      limx->-2f(x)   limx->1-f(x)   limx->1+f(x)    F(1)  answer all

To answer the provided limit and function evaluation questions, let's carefully analyze the graph:

1. **\(\lim_{x \to -2^-} f(x)\)**: This asks for the limit of the function as \(x\) approaches \(-2\) from the left-hand side. Looking at the graph near \(x = -2\) from the left, we see the function approaching a value of **3**.
   - **Answer:** \(\lim_{x \to -2^-} f(x) = 3\).

2. **\(\lim_{x \to -2^+} f(x)\)**: This asks for the limit of the function as \(x\) approaches \(-2\) from the right-hand side. From the right side of \(x = -2\), the function approaches a value of **2**.
   - **Answer:** \(\lim_{x \to -2^+} f(x) = 2\).

3. **\(f(-2)\)**: The function value at \(x = -2\) is represented by a **filled circle** on the graph at the point \( (-2, 3) \), meaning \(f(-2) = 3\).
   - **Answer:** \(f(-2) = 3\).

4. **\(\lim_{x \to -2} f(x)\)**: Since the left-hand limit (\( \lim_{x \to -2^-} f(x) = 3 \)) and the right-hand limit (\( \lim_{x \to -2^+} f(x) = 2 \)) are **not equal**, the overall limit does not exist.
   - **Answer:** \(\lim_{x \to -2} f(x)\) does not exist.

5. **\(\lim_{x \to 1^-} f(x)\)**: As \(x\) approaches 1 from the left-hand side, the graph approaches a value of **-4**.
   - **Answer:** \(\lim_{x \to 1^-} f(x) = -4\).

6. **\(\lim_{x \to 1^+} f(x)\)**: As \(x\) approaches 1 from the right-hand side, the graph approaches a value of **4**.
   - **Answer:** \(\lim_{x \to 1^+} f(x) = 4\).

7. **\(f(1)\)**: The value of the function at \(x = 1\) is indicated by an **open circle** at \( (-4) \), meaning the function is not defined at \(x = 1\).
   - **Answer:** \(f(1)\) is undefined.

### Summary of Answers:
1. \(\lim_{x \to -2^-} f(x) = 3\)
2. \(\lim_{x \to -2^+} f(x) = 2\)
3. \(f(-2) = 3\)
4. \(\lim_{x \to -2} f(x)\) does not exist
5. \(\lim_{x \to 1^-} f(x) = -4\)
6. \(\lim_{x \to 1^+} f(x) = 4\)
7. \(f(1)\) is undefined

Would you like any of these points explained further?

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Here are 5 related questions for deeper exploration:
1. What happens to the limit when the left and right limits are equal?
2. How can we determine if a function is continuous at a certain point based on limits?
3. What would be the behavior of the function if it had a filled circle at \(x = 1\)?
4. Can you have a function that is defined at \(x = 1\) but the limit at \(x = 1\) does not exist?
5. How would you describe the overall continuity of this graph?

### Tip:
A limit exists at a point only if both the left-hand and right-hand limits are equal at that point.

answer all of the questions below limx->-2-f(x) limx->-2+f(x) f(-2) limx->-2f(x) limx->1-f(x) limx->1+f(x) F(1) answer all

Analyze limits and function values using a piecewise function graph. Questions include evaluating left-hand and right-hand limits at x = -2 and x = 1, checking for continuity, and determining function values based on filled and open circles.

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