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

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