I don't know how to prove that $\lim_{x\to\infty}\frac{\ln x}{x-1}=0$ without using L'Hopital.
I've tried to use the definition of $\ln x=\int_{1}^{x}\frac{1}{t}dt$ and the fact that $\frac{x-1}{x}<\ln x<x-1$ but I don't get anything. Another thing I've tried is to prove the limit by definition but it's quite complicated. Any suggestions?