The task involves analyzing the provided graph and finding several limits of the function $f(x)$, as well as evaluating some specific function values. Let me go through each one:

1. $\lim_{x \to \infty} f(x)$:

   • As $x$ approaches infinity, the function seems to approach the horizontal asymptote at $y = 2$.
   • Answer: $2$

2. $\lim_{x \to -\infty} f(x)$:

   • As $x$ approaches negative infinity, the function approaches a horizontal asymptote at $y = -4$.
   • Answer: $-4$

3. $\lim_{x \to -2} f(x)$:

   • As $x$ approaches $-2$ from both sides, the function value heads towards $-3$ (there is a hole at $(-2, -3)$).
   • Answer: $-3$

4. $f(-2)$:

   • Since there is a hole at $(-2, -3)$, the function is not defined at $x = -2$.
   • Answer: Undefined

5. $\lim_{x \to -1^-} f(x)$:

   • As $x$ approaches $-1$ from the left, the function goes to $-\infty$ (the vertical asymptote).
   • Answer: $-\infty$

6. $\lim_{x \to -1^+} f(x)$:

   • As $x$ approaches $-1$ from the right, the function goes to $\infty$.
   • Answer: $\infty$

7. $\lim_{x \to -1} f(x)$:

   • Since the left-hand and right-hand limits do not match (one goes to $-\infty$ and the other goes to $\infty$), the limit does not exist.
   • Answer: DNE (Does Not Exist)

8. $f(-1)$:

   • At $x = -1$, there is a vertical asymptote, so the function is not defined.
   • Answer: Undefined

9. $\lim_{x \to 0^-} f(x)$:

   • As $x$ approaches $0$ from the left, the function approaches $0$.
   • Answer: $0$

10. $\lim_{x \to 0^+} f(x)$:

    • As $x$ approaches $0$ from the right, the function also approaches $0$.
    • Answer: $0$

11. $\lim_{x \to 0} f(x)$:

    • Since both the left-hand and right-hand limits are equal to $0$, the overall limit exists and equals $0$.
    • Answer: $0$

12. $f(0)$:

    • From the graph, $f(0) = 0$.
    • Answer: $0$

13. $\lim_{x \to 2^-} f(x)$:

    • As $x$ approaches $2$ from the left, the function approaches $2$.
    • Answer: $2$

14. $\lim_{x \to 2^+} f(x)$:

    • As $x$ approaches $2$ from the right, the function approaches $4$.
    • Answer: $4$

15. $\lim_{x \to 2} f(x)$:

    • Since the left-hand and right-hand limits are different ($2$ and $4$), the overall limit does not exist.
    • Answer: DNE (Does Not Exist)

16. $f(2)$:

    • From the graph, $f(2) = 4$.
    • Answer: $4$

Would you like further clarification on any of the limits or more details?

Here are some related questions to explore:

1. What is the formal definition of a limit, and how does it apply to discontinuous functions?
2. How do vertical asymptotes affect the limits of a function?
3. What is the difference between left-hand and right-hand limits?
4. Why does a function's limit not exist at points with differing left-hand and right-hand limits?
5. How do you identify the value of a limit graphically?

Tip: Always check if the function is defined at a specific point, as this can differ from the limit approaching that point.