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The problem you uploaded appears to be analyzing the **asymptotic behavior of a rational function**. Based on the graphs provided, we are likely focusing on identifying how the function behaves as \( x \) approaches infinity, negative infinity, or a vertical asymptote.

### Steps to solve:

1. **Identify horizontal asymptote**:
   - As \( x \to \infty \), the graph flattens, suggesting a horizontal asymptote.
   - The same occurs as \( x \to -\infty \).

2. **Identify vertical asymptote**:
   - There is a vertical asymptote at \( x = 3 \) since the graph approaches \( \pm \infty \) near \( x = 3 \).

3. **Analyze left-hand and right-hand limits near the asymptote**:
   - As \( x \to 3^+ \) (from the right), the function approaches \( \infty \).
   - As \( x \to 3^- \) (from the left), the function approaches \( -\infty \).

Now, let's look at the statements provided in the second image and compare them with the graph:

1. **As \( x \to -\infty \), \( f(x) \to -2 \)**:
   - This seems consistent with the graph. The horizontal asymptote on the left side approaches \( y = -2 \).

2. **As \( x \to 3^+ \), \( f(x) \to \infty \)**:
   - This is correct. The graph shoots upwards near \( x = 3 \) from the right side.

3. **As \( x \to \infty \), \( f(x) \to -2 \)**:
   - This is incorrect. As \( x \to \infty \), the graph seems to approach \( y = 2 \), not \( -2 \).

4. **As \( x \to 3^-\), \( f(x) \to -\infty \)**:
   - This is correct. From the left side of \( x = 3 \), the function heads towards \( -\infty \).

5. **As \( x \to \infty \), \( f(x) \to 2 \)**:
   - This is correct. The horizontal asymptote on the right side approaches \( y = 2 \).

### Correct answers:
- The valid statements are:
   - As \( x \to -\infty, f(x) \to -2 \)
   - As \( x \to 3^+, f(x) \to \infty \)
   - As \( x \to 3^-, f(x) \to -\infty \)
   - As \( x \to \infty, f(x) \to 2 \)

Would you like further details on the reasoning for any part?

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Here are some related questions to explore:

1. What does a horizontal asymptote tell us about a rational function’s behavior at infinity?
2. How do vertical asymptotes affect the local behavior of rational functions?
3. How can we determine the horizontal asymptote of a rational function algebraically?
4. What is the significance of left-hand and right-hand limits near an asymptote?
5. How does the degree of the numerator and denominator in a rational function influence its asymptotes?

**Tip:** Always check the end behavior by evaluating limits as \( x \to \infty \) and \( x \to -\infty \) to confirm horizontal asymptotes.

The graph of the rational function f(x) is shown below. Using the graph, determine which of the following local and end behaviors are correct.

Learn how to analyze the asymptotic behavior of rational functions by identifying end behaviors, horizontal asymptotes, and vertical asymptotes. This problem involves calculating limits as x approaches infinity and determining the function's behavior around its asymptotes.

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