https://file.aiquickdraw.com/mathbot/file/17266290375916ivhmjax.jpg

The image shows a graph of a function \( F(x) \) and several questions about limits and function values based on that graph.

### Evaluating Limits and Function Values

Let's analyze the graph to answer the expressions:

#### (a) 
1. \( \lim_{x \to -1^-} F(x) \):
   - As \( x \) approaches \(-1\) from the left, the value of the function approaches \( F(-1^-) = 0 \).

2. \( \lim_{x \to -1^+} F(x) \):
   - As \( x \) approaches \(-1\) from the right, the value of the function approaches \( F(-1^+) = 3 \).

3. Since \( \lim_{x \to -1^-} \neq \lim_{x \to -1^+} \), the two-sided limit \( \lim_{x \to -1} F(x) \) does not exist (DNE).

4. The value of \( F(-1) \) is given directly as \( F(-1) = 3 \).

#### (b)
1. \( \lim_{x \to 1^-} F(x) \):
   - As \( x \) approaches 1 from the left, the function approaches \( F(1^-) = 3 \).

2. \( \lim_{x \to 1^+} F(x) \):
   - As \( x \) approaches 1 from the right, the function approaches \( F(1^+) = 1 \).

3. Since \( \lim_{x \to 1^-} \neq \lim_{x \to 1^+} \), the two-sided limit \( \lim_{x \to 1} F(x) \) does not exist (DNE).

4. The value of \( F(1) \) is \( F(1) = 1 \).

#### (c)
1. \( \lim_{x \to 3^-} F(x) \):
   - As \( x \) approaches 3 from the left, the function approaches \( F(3^-) = -3 \).

2. \( \lim_{x \to 3^+} F(x) \):
   - As \( x \) approaches 3 from the right, the function approaches \( F(3^+) = -3 \).

3. Since \( \lim_{x \to 3^-} = \lim_{x \to 3^+} \), the two-sided limit \( \lim_{x \to 3} F(x) \) exists and is equal to \(-3\).

4. The value of \( F(3) \) is not defined, so \( F(3) = \text{DNE} \).

### Conclusion for each question:

1. \( \lim_{x \to -1^-} F(x) = 0 \)
2. \( \lim_{x \to -1^+} F(x) = 3 \)
3. \( \lim_{x \to -1} F(x) = \text{DNE} \)
4. \( F(-1) = 3 \)
---
1. \( \lim_{x \to 1^-} F(x) = 3 \)
2. \( \lim_{x \to 1^+} F(x) = 1 \)
3. \( \lim_{x \to 1} F(x) = \text{DNE} \)
4. \( F(1) = 1 \)
---
1. \( \lim_{x \to 3^-} F(x) = -3 \)
2. \( \lim_{x \to 3^+} F(x) = -3 \)
3. \( \lim_{x \to 3} F(x) = -3 \)
4. \( F(3) = \text{DNE} \)

Would you like further details on any of these limits or function evaluations? Here are some questions you could explore next:

1. What is the difference between one-sided and two-sided limits?
2. Why does the limit not exist at \( x = -1 \) and \( x = 1 \)?
3. How does continuity relate to the limits at \( x = 3 \)?
4. What happens to the function behavior at other points like \( x = 0 \) or \( x = 4 \)?
5. Can the graph tell us more about differentiability at \( x = -1 \), \( x = 1 \), or \( x = 3 \)?

Tip: When evaluating limits, always check if the left-hand and right-hand limits agree to determine if the two-sided limit exists!

Evaluate the following limits and function values based on the graph of the function F(x).

This detailed guide helps students understand how to evaluate limits and function values for a piecewise function graphically. It covers key concepts such as one-sided limits, the condition for the existence of two-sided limits, and function evaluation techniques.

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