To answer the provided limit and function evaluation questions, let's carefully analyze the graph:

1. $\lim_{x \to -2^-} f(x)$: This asks for the limit of the function as $x$ approaches $-2$ from the left-hand side. Looking at the graph near $x = -2$ from the left, we see the function approaching a value of 3.

   • Answer: $\lim_{x \to -2^-} f(x) = 3$.

2. $\lim_{x \to -2^+} f(x)$: This asks for the limit of the function as $x$ approaches $-2$ from the right-hand side. From the right side of $x = -2$, the function approaches a value of 2.

   • Answer: $\lim_{x \to -2^+} f(x) = 2$.

3. $f(-2)$: The function value at $x = -2$ is represented by a filled circle on the graph at the point $(-2, 3)$, meaning $f(-2) = 3$.

   • Answer: $f(-2) = 3$.

4. $\lim_{x \to -2} f(x)$: Since the left-hand limit ($\lim_{x \to -2^-} f(x) = 3$) and the right-hand limit ($\lim_{x \to -2^+} f(x) = 2$) are not equal, the overall limit does not exist.

   • Answer: $\lim_{x \to -2} f(x)$ does not exist.

5. $\lim_{x \to 1^-} f(x)$: As $x$ approaches 1 from the left-hand side, the graph approaches a value of -4.

   • Answer: $\lim_{x \to 1^-} f(x) = -4$.

6. $\lim_{x \to 1^+} f(x)$: As $x$ approaches 1 from the right-hand side, the graph approaches a value of 4.

   • Answer: $\lim_{x \to 1^+} f(x) = 4$.

7. $f(1)$: The value of the function at $x = 1$ is indicated by an open circle at $(-4)$, meaning the function is not defined at $x = 1$.

   • Answer: $f(1)$ is undefined.

Summary of Answers:

1. $\lim_{x \to -2^-} f(x) = 3$
2. $\lim_{x \to -2^+} f(x) = 2$
3. $f(-2) = 3$
4. $\lim_{x \to -2} f(x)$ does not exist
5. $\lim_{x \to 1^-} f(x) = -4$
6. $\lim_{x \to 1^+} f(x) = 4$
7. $f(1)$ is undefined

Would you like any of these points explained further?

Here are 5 related questions for deeper exploration:

1. What happens to the limit when the left and right limits are equal?
2. How can we determine if a function is continuous at a certain point based on limits?
3. What would be the behavior of the function if it had a filled circle at $x = 1$?
4. Can you have a function that is defined at $x = 1$ but the limit at $x = 1$ does not exist?
5. How would you describe the overall continuity of this graph?

Tip:

A limit exists at a point only if both the left-hand and right-hand limits are equal at that point.