We need to evaluate the limit:

$\lim_{x \to +\infty} \frac{-3x^3 - 2x + 1}{2x^3 - 5x + 2}$

Step 1: Identify the highest degree term

The highest degree term in both the numerator and denominator is $x^3$. To simplify, we factor $x^3$ out of both the numerator and denominator:

$\frac{-3x^3 - 2x + 1}{2x^3 - 5x + 2} = \frac{x^3(-3 - \frac{2}{x^2} + \frac{1}{x^3})}{x^3(2 - \frac{5}{x^2} + \frac{2}{x^3})}$

Step 2: Evaluate the limit

As $x \to +\infty$, the terms $\frac{2}{x^2}$, $\frac{1}{x^3}$, $\frac{5}{x^2}$, and $\frac{2}{x^3}$ approach 0. So the expression simplifies to:

$\frac{-3}{2}$

Conclusion:

$\lim_{x \to +\infty} \frac{-3x^3 - 2x + 1}{2x^3 - 5x + 2} = -\frac{3}{2}$

Would you like a more detailed explanation? Here are some related questions:

1. How does the limit change if the highest degree terms are different?
2. What if the leading coefficients in the numerator and denominator are the same?
3. How can L'Hôpital's Rule be used for polynomial limits?
4. What happens if the degree of the numerator is greater than the denominator?
5. How can we use asymptotic analysis to approximate rational function limits?

Tip: The behavior of rational functions at infinity is determined by the highest-degree terms!