1. **State the problem:** We want to find the limit $$\lim_{n \to \infty} n \ln\left(1 - \frac{5}{n}\right).$$ 2. **Rewrite the expression inside the logarithm:** As $n$ becomes very large, $\frac{5}{n}$ approaches 0, so the expression inside the logarithm approaches 1. 3. **Use the approximation for logarithm near 1:** For small $x$, $\ln(1+x) \approx x - \frac{x^2}{2} + \cdots$. Here, $x = -\frac{5}{n}$, which is small when $n$ is large. 4. **Apply the approximation:** $$\ln\left(1 - \frac{5}{n}\right) \approx -\frac{5}{n} - \frac{\left(-\frac{5}{n}\right)^2}{2} = -\frac{5}{n} - \frac{25}{2n^2}.$$ 5. **Multiply by $n$:** $$n \ln\left(1 - \frac{5}{n}\right) \approx n \left(-\frac{5}{n} - \frac{25}{2n^2}\right) = -5 - \frac{25}{2n}.$$ 6. **Take the limit as $n \to \infty$:** The term $\frac{25}{2n}$ goes to 0, so $$\lim_{n \to \infty} n \ln\left(1 - \frac{5}{n}\right) = -5.$$ **Final answer:** $$\boxed{-5}.$$