1. **State the problem:** Find the limit $$\lim_{n \to \infty} \left(\frac{2n + 10}{3n + 2}\right)^n.$$\n\n2. **Rewrite the expression inside the limit:** Simplify the fraction inside the parentheses by dividing numerator and denominator by $n$:\n$$\left(\frac{2n + 10}{3n + 2}\right)^n = \left(\frac{2 + \frac{10}{n}}{3 + \frac{2}{n}}\right)^n.$$\n\n3. **Evaluate the limit of the base as $n \to \infty$:** Since $\frac{10}{n} \to 0$ and $\frac{2}{n} \to 0$, the base approaches $$\frac{2 + 0}{3 + 0} = \frac{2}{3}.$$\n\n4. **Analyze the limit:** The expression is of the form $$\left(\text{something less than 1}\right)^n$$ as $n \to \infty$. Since $\frac{2}{3} < 1$, raising it to the power $n$ where $n$ grows without bound will make the expression approach zero.\n\n5. **Conclusion:** Therefore, $$\lim_{n \to \infty} \left(\frac{2n + 10}{3n + 2}\right)^n = 0.$$