convergence divergence - Seeking a rigorous proof for a limit

Prove that$$\lim_{n\to\infty}\frac{\ln (1- \frac{3}{n})}{n}=0$$

I knew that this is the indeterminate form $0/\infty$(Actually it isn't, so I made a dumb mistake) that should be zero but was unable to prove it. I haven't tried using the definition yet because I feel that's too cumbersome?

Answer

Note that the limit is not an indeterminate form, indeed:

$$\ln \left(1- \frac{3}{n}\right)\to \ln 1=0 \implies \frac{\ln (1- \frac{3}{n})}{n}\to\frac{0}{+\infty}=0$$