1. **State the problem:** Find the limit $$\lim_{x \to \infty} 2x \ln\left(1 + \frac{5}{x}\right).$$\n\n2. **Recall the formula and rule:** For limits involving expressions like $x \ln(1 + \frac{a}{x})$ as $x \to \infty$, we use the fact that $\ln(1 + y) \approx y$ when $y \to 0$. Here, $y = \frac{5}{x} \to 0$ as $x \to \infty$.\n\n3. **Rewrite the expression:**\n$$2x \ln\left(1 + \frac{5}{x}\right) = 2x \cdot \ln\left(1 + \frac{5}{x}\right).$$\n\n4. **Use substitution:** Let $y = \frac{1}{x}$, so as $x \to \infty$, $y \to 0^+$. The expression becomes\n$$2 \cdot \frac{1}{y} \cdot \ln(1 + 5y).$$\n\n5. **Apply the approximation for small $y$:**\n$$\ln(1 + 5y) \approx 5y.$$\n\n6. **Substitute back:**\n$$2 \cdot \frac{1}{y} \cdot 5y = 2 \cdot \frac{1}{y} \cdot 5y = 2 \cdot 5 = 10.$$\n\n7. **Conclusion:** The limit is\n$$\boxed{10}.$$