1. **State the problem:** Find the limit as $x$ approaches infinity of the function $$\lim_{x \to \infty} \frac{5x^2 - 7}{x^2 - 5x}.$$\n\n2. **Recall the rule for limits at infinity of rational functions:** When the degrees of the numerator and denominator polynomials are the same, the limit is the ratio of the leading coefficients.\n\n3. **Identify the leading terms:** The highest power of $x$ in numerator and denominator is $x^2$. The leading coefficient in the numerator is 5, and in the denominator is 1.\n\n4. **Divide numerator and denominator by $x^2$ to simplify:**\n$$\lim_{x \to \infty} \frac{5x^2 - 7}{x^2 - 5x} = \lim_{x \to \infty} \frac{\frac{5x^2}{x^2} - \frac{7}{x^2}}{\frac{x^2}{x^2} - \frac{5x}{x^2}} = \lim_{x \to \infty} \frac{5 - \frac{7}{x^2}}{1 - \frac{5}{x}}.$$\n\n5. **Evaluate the limit by substituting $x \to \infty$:**\nSince $\lim_{x \to \infty} \frac{7}{x^2} = 0$ and $\lim_{x \to \infty} \frac{5}{x} = 0$, we get\n$$\frac{5 - 0}{1 - 0} = \frac{5}{1} = 5.$$\n\n**Final answer:** $$\boxed{5}.$$