Finding limit without using L'Hopital rule How to prove $$\lim_{x \to \infty} \frac {\ln(x)}{x}=0$$ without using L'Hospital Rule. Just by using some basis limit properties.
For sufficiently large $x$:$0\le \ln (x)\le \sqrt x$. Now divide by $x$ and squeeze.
The Taylor series for $\ln(x)$ in $-1\le x \le 1$ is given by : 



You can substitute this and then apply the basic limits results to get the answer.