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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.

Find all of the limits of the following graph.

Learn how to find limits and evaluate function values based on the graph of a function, including behavior at asymptotes and discontinuities. This covers concepts of limits, left-hand and right-hand limits, and function behavior at infinity.

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