Limit $lim_{xtoinfty} frac {(ln(x))^k}x$

The limit is

$$\lim_{x\to\infty} \frac {(\ln(x))^k}x$$ $\forall k \in N, k \ge 1$

For $k = 1$ or $2$, the limit is fairly easy to solve using the Stolz–Cesàro theorem. Intuitively, because $ln(x)$ grows so much slower than $x$, the answer should be $0$. But how can we mathematically solve the limit for any positive integer k?

Answer

$$\lim_{x\to\infty}\frac{\ln^kx}{x}= \left(\lim_{x\to\infty}\frac{k\ln x^{\frac1{k}}}{x^{\frac1{k}}} \right)^k = k^k \left( \lim_{y\to\infty} \frac{\ln y}{y} \right)^k= 0$$