limits - Evaluation $lim_{nto infty}frac{{log^k n}}{n^{epsilon}}$

Evaluate where $\epsilon>0,k\geqslant 1$ are constants

$$\lim_{n\to \infty}\frac{{\log^k n}}{n^{\epsilon}}$$

L'Hopital can't help here, also I tried to use $\log$ rules but it didn't helped, I know that $\log$ grows slower then polynom, but $n^\epsilon$ is not polynom, how can I evaluate this limit? thank you

Answer

Write

$$\frac{(\log n)^k}{n^\varepsilon}= \left(\frac{\log n}{n^{\varepsilon/k}}\right)^{\!k}$$
For $r>0$, we have

$$\lim_{x\to\infty}\frac{\log x}{x^r}= \lim_{x\to\infty}\frac{1/x}{rx^{r-1}}= \lim_{x\to\infty}\frac{1}{rx^r}=0$$