https://file.aiquickdraw.com/mathbot/file/1725887520355gtpw2snz.jpg

Let's evaluate each of the given expressions using the graph provided:

### a) \(\lim_{x \to -1} F(x)\)

To find the limit as \(x\) approaches \(-1\), we look at the values of the function from both the left and the right:

- From the left (\(x \to -1^-\)), \(F(x)\) is approaching 3.
- From the right (\(x \to -1^+\)), \(F(x)\) is also approaching 3.

Since both the left and right limits are equal, the limit exists and is:

\[
\lim_{x \to -1} F(x) = 3
\]

### b) \(\lim_{x \to 1^+} F(x)\)

This is the limit as \(x\) approaches \(1\) from the right:

- From the right (\(x \to 1^+\)), the function value is approaching \(4\).

Thus:

\[
\lim_{x \to 1^+} F(x) = 4
\]

### c) \(\lim_{x \to -1} F(x)\)

This appears to be a repetition of part (a), so:

\[
\lim_{x \to -1} F(x) = 3
\]

### d) \(F(-1)\)

Given that \(F(-1) = 3\).

### e) \(\lim_{x \to -1^+} F(x)\)

This is the limit as \(x\) approaches \(-1\) from the right:

- From the right (\(x \to -1^+\)), the function value is approaching 3.

So:

\[
\lim_{x \to -1^+} F(x) = 3
\]

### f) \(\lim_{x \to 1^-} F(x)\)

This is the limit as \(x\) approaches \(1\) from the left:

- From the left (\(x \to 1^-\)), the function value is approaching \(2\).

Therefore:

\[
\lim_{x \to 1^-} F(x) = 2
\]

### g) \(\lim_{x \to 1} F(x)\)

Since the left-hand limit \(\lim_{x \to 1^-} F(x) = 2\) and the right-hand limit \(\lim_{x \to 1^+} F(x) = 4\), the limit does not exist:

\[
\lim_{x \to 1} F(x) = \text{DNE}
\]

### h) \(\lim_{x \to 3} F(x)\)

Approaching \(x = 3\) from both sides:

- From the left, \(F(x)\) is approaching \(0\).
- From the right, \(F(x)\) is approaching \(0\).

Thus:

\[
\lim_{x \to 3} F(x) = 0
\]

### i) \(F(3)\)

At \(x = 3\), it appears there is no defined point for \(F(3)\), so:

\[
F(3) = \text{DNE}
\]

Let me know if you have any questions or would like further details!

### Further Questions:
1. What is the significance of a limit existing at a point even if the function is not defined at that point?
2. How do you determine the left-hand and right-hand limits from a graph?
3. What does it mean if the left-hand limit and right-hand limit at a point are different?
4. How can limits help in understanding the continuity of a function?
5. How would you describe the overall behavior of the function near \(x = 1\)?

**Tip:** When examining limits on a graph,

Evaluate the following expressions using the graph provided for the function F(x): a) lim(x→−1) F(x), b) lim(x→1+) F(x), c) lim(x→−1) F(x), d) F(−1), e) lim(x→−1+) F(x), f) lim(x→1−) F(x), g) lim(x→1) F(x), h) lim(x→3) F(x), i) F(3).

Learn how to evaluate limits and determine continuity for a given function using graph analysis. This problem covers key concepts such as left-hand limits, right-hand limits, and the existence of limits, suitable for Grades 11-12 students.

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