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!