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,