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?

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.