https://file.aiquickdraw.com/mathbot/file/1726847702309mmr4f432.jpg

The polynomial given is \( f(x) = -2x^5(6x - 9)^1(x^2 + 1)^2 \).

To determine the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \), we will analyze the leading term since it dominates the behavior at extreme values of \( x \).

### Leading Term Analysis:
- The highest degree term in \( f(x) \) is dominated by \( -2x^5 \), as it will grow the fastest when \( x \) becomes very large or very small (positive or negative infinity).

#### As \( x \to \infty \):
- \( x^5 \to \infty \), and since there is a negative coefficient (-2), the entire expression tends to \( -\infty \).

#### As \( x \to -\infty \):
- \( (-x)^5 = -x^5 \), so \( x^5 \to -\infty \), and multiplying by the negative coefficient (-2) results in \( f(x) \to \infty \).

### Conclusion:
- \( \lim_{x \to \infty} f(x) = -\infty \)
- \( \lim_{x \to -\infty} f(x) = \infty \)

The correct statement is:
- \( \lim_{x \to \infty} f(x) = -\infty \) and \( \lim_{x \to -\infty} f(x) = \infty \), which corresponds to the **third option** in the image.

Would you like further details or have any questions? Here are some related questions to explore:

1. How do we determine the dominant term in a polynomial function?
2. What is the impact of negative coefficients on the limits of polynomials?
3. Why does the degree of a polynomial determine its end behavior?
4. How would the behavior of \( f(x) \) change if the leading term had an even degree?
5. Can we generalize the end behavior of all polynomials based on their leading term?

**Tip**: Always focus on the highest-degree term when analyzing the behavior of polynomials as \( x \to \pm \infty \).

The polynomial is given by f(x) = -2x^5(6x - 9)(x^2 + 1)^2. Which of the following is a true statement about the function f?

Understand how to find the limits of the polynomial function f(x) = -2x^5(6x - 9)(x^2 + 1)^2 as x approaches infinity and negative infinity. Learn the key concepts behind polynomial end behavior and asymptotic analysis, suitable for high school students.

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