1. **State the problem:** Find the limit $$\lim_{x \to +\infty} \frac{2x-3}{x^2+x+1}$$. 2. **Recall the rule for limits at infinity:** When evaluating limits of rational functions as $x$ approaches infinity, compare the degrees of the numerator and denominator. 3. **Identify degrees:** The numerator $2x-3$ is degree 1, and the denominator $x^2+x+1$ is degree 2. 4. **Since the degree of the denominator is higher, the limit tends to zero.** 5. **To confirm, divide numerator and denominator by $x^2$ (the highest power in denominator):** $$\lim_{x \to +\infty} \frac{\frac{2x}{x^2} - \frac{3}{x^2}}{\frac{x^2}{x^2} + \frac{x}{x^2} + \frac{1}{x^2}} = \lim_{x \to +\infty} \frac{\frac{2}{x} - \frac{3}{x^2}}{1 + \frac{1}{x} + \frac{1}{x^2}}$$ 6. **Evaluate the limit by substituting $x \to +\infty$: all terms with $\frac{1}{x}$ or $\frac{1}{x^2}$ go to zero:** $$\frac{0 - 0}{1 + 0 + 0} = 0$$ **Final answer:** $$\boxed{0}$$