1. **State the problem:** Find the limit $$\lim_{x \to \infty} \frac{2x^3 - 5x + 1}{4x^2 + 3x - 2}$$. 2. **Recall the rule for limits at infinity of rational functions:** When $x \to \infty$, the behavior of the function is dominated by the highest degree terms in numerator and denominator. 3. **Identify the highest degree terms:** Numerator highest degree term is $2x^3$, denominator highest degree term is $4x^2$. 4. **Divide numerator and denominator by $x^2$ (the highest power in denominator):** $$\frac{2x^3 - 5x + 1}{4x^2 + 3x - 2} = \frac{\frac{2x^3}{x^2} - \frac{5x}{x^2} + \frac{1}{x^2}}{\frac{4x^2}{x^2} + \frac{3x}{x^2} - \frac{2}{x^2}} = \frac{2x - \frac{5}{x} + \frac{1}{x^2}}{4 + \frac{3}{x} - \frac{2}{x^2}}$$ 5. **Evaluate the limit as $x \to \infty$:** Terms with $\frac{1}{x}$ and $\frac{1}{x^2}$ go to zero, so $$\lim_{x \to \infty} \frac{2x - 0 + 0}{4 + 0 - 0} = \lim_{x \to \infty} \frac{2x}{4} = \lim_{x \to \infty} \frac{x}{2} = \infty$$ 6. **Conclusion:** The limit diverges to infinity. **Final answer:** $$\lim_{x \to \infty} \frac{2x^3 - 5x + 1}{4x^2 + 3x - 2} = \infty$$