It looks like you've uploaded an image with a piece of math homework about limits and a piecewise function. I see the graph of the function and the questions related to evaluating the limits and the function at specific points.

From what I can make out, you are asked to evaluate the following:

1. $\lim_{x \to -1^-} F(x)$
2. $\lim_{x \to -1^+} F(x)$
3. $\lim_{x \to -1} F(x)$
4. $\lim_{x \to 1^-} F(x)$
5. $\lim_{x \to 1^+} F(x)$
6. $\lim_{x \to 1} F(x)$
7. $\lim_{x \to 3^-} F(x)$
8. $\lim_{x \to 3^+} F(x)$
9. $F(3)$

To proceed, let's analyze the graph:

• a) and b) ask for the left-hand limit ($x \to -1^-$) and right-hand limit ($x \to -1^+$) at $x = -1$.
• c) asks for the overall limit at $x = -1$.
• d) asks for the limit as $x$ approaches 1 from the left.
• e) asks for the limit as $x$ approaches 1 from the right.
• f) asks for the overall limit at $x = 1$.
• g) and h) ask for the limits as $x$ approaches 3 from the left and right.
• i) asks for the value of the function at $x = 3$.

I'll need to carefully analyze the graph you uploaded to give you precise values for each of these. Let me take a closer look at the graph and provide the answers.

Give me a moment.I’ve analyzed the graph, and here’s a breakdown of the limits and function values based on the points shown:

1. a) $\lim_{x \to -1^-} F(x)$: This represents the left-hand limit at $x = -1$. Looking at the graph, the value of the function as $x$ approaches -1 from the left is approximately 1.

2. b) $\lim_{x \to -1^+} F(x)$: This represents the right-hand limit at $x = -1$. From the graph, as $x$ approaches -1 from the right, the value is -1.

3. c) $\lim_{x \to -1} F(x)$: For the overall limit to exist, the left and right limits must be the same. Since they are different (1 and -1), the overall limit at $x = -1$ does not exist.

4. d) $\lim_{x \to 1^-} F(x)$: This is the left-hand limit at $x = 1$. From the graph, as $x$ approaches 1 from the left, the value of the function is -2.

5. e) $\lim_{x \to 1^+} F(x)$: This is the right-hand limit at $x = 1$. From the graph, as $x$ approaches 1 from the right, the value is 2.

6. f) $\lim_{x \to 1} F(x)$: Since the left and right limits at $x = 1$ are different (-2 and 2), the overall limit at $x = 1$ does not exist.

7. g) $\lim_{x \to 3^-} F(x)$: This is the left-hand limit at $x = 3$. As $x$ approaches 3 from the left, the value is 1.

8. h) $\lim_{x \to 3^+} F(x)$: This is the right-hand limit at $x = 3$. As $x$ approaches 3 from the right, the value is 3.

9. i) $F(3)$: The value of the function at $x = 3$ is 2, as shown by the solid dot on the graph at this point.

If you need further clarification or details about these steps, feel free to ask!

Here are some follow-up questions to help solidify your understanding:

1. What is the difference between left-hand limits and right-hand limits?
2. How do you determine if a function has a limit at a point where the left and right limits differ?
3. What do discontinuities in a graph tell you about the behavior of the function at a particular point?
4. How would the behavior of the function change if the limits at a point are equal, but there’s a jump discontinuity?
5. How would you approach calculating limits for piecewise functions with more complicated expressions?

Tip: Always check if the left-hand and right-hand limits match when determining if the overall limit exists at a point!