1. **State the problem:** Find the limit as $n$ approaches infinity of the expression $$\frac{3n^2 + 5}{2n^3 - 7n + 5}.$$\n\n2. **Analyze the degrees of polynomials:** The numerator is a polynomial of degree 2 ($3n^2$ is the highest degree term). The denominator is a polynomial of degree 3 ($2n^3$ is the highest degree term).\n\n3. **Divide numerator and denominator by $n^3$ (the highest power in denominator):**\n$$\frac{\frac{3n^2}{n^3} + \frac{5}{n^3}}{\frac{2n^3}{n^3} - \frac{7n}{n^3} + \frac{5}{n^3}} = \frac{\frac{3}{n} + \frac{5}{n^3}}{2 - \frac{7}{n^2} + \frac{5}{n^3}}.$$\n\n4. **Take the limit as $n \to \infty$: all terms with $\frac{1}{n^k}$ where $k > 0$ go to 0:**\n$$\lim_{n \to \infty} \frac{\frac{3}{n} + \frac{5}{n^3}}{2 - \frac{7}{n^2} + \frac{5}{n^3}} = \frac{0 + 0}{2 - 0 + 0} = \frac{0}{2} = 0.$$\n\n5. **Conclusion:** The limit is 0.