I'm trying to prove or disprove that $\log^{k}(n) \in O(\sqrt{n}), \ \forall k > 0$. By using the free version of wolfram and testing some increasing values of $k$ I get that:
$$\lim_{n \rightarrow \infty} \frac{\log^{k}(n)}{\sqrt{n}} = 0$$
And apparently $\log^{k}(n) \in O(\sqrt{n})$, but by trying to solve this limit on paper in order to reach an appropriate proof I would need to keep applying L'Hospital rule indefinitely. Is that what I suppose to be doing? How could I proceed to build this proof?