How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$ How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$

Let $A=\lim_{n\to\infty} \frac{(\log n)^p}{n}$.Taking $log$ both sides gives $\log A=\lim_{n\to\infty} (p\log \log n -\log n)=\lim_{\log n\to\infty} (p\log \log n -\log n) $.Take $y=1/\log n\implies \log A=\lim_{y\to0^+} (-p\log y -1/y) \implies \log A=\lim_{y\to0^+} \frac{-py\log y -1}{y}$.This limit surely goes to $-\infty$ and hence $\log A=-\infty\implies A=0$.Thus the limit is $0$.
