1. **State the problem:** Find the limit as $x \to \infty$ of the function $$\frac{x - 25}{x^2 + 1}$$. 2. **Recall the rule for limits of rational functions at infinity:** When the degree of the denominator is higher than the degree of the numerator, the limit is 0. 3. **Apply the rule:** The numerator is degree 1, the denominator is degree 2. 4. **Divide numerator and denominator by $x^2$ (highest power in denominator):** $$\frac{\frac{x}{x^2} - \frac{25}{x^2}}{\frac{x^2}{x^2} + \frac{1}{x^2}} = \frac{\frac{1}{x} - \frac{25}{x^2}}{1 + \frac{1}{x^2}}$$ 5. **Evaluate the limit as $x \to \infty$:** $$\lim_{x \to \infty} \frac{\frac{1}{x} - \frac{25}{x^2}}{1 + \frac{1}{x^2}} = \frac{0 - 0}{1 + 0} = 0$$ 6. **Final answer:** $$\boxed{0}$$