Trying to understand a proof

Question

For $x \in (0,1]$, Since $$\lim\limits_{x\to0} \frac{\Big|\ln\frac{1}{x}\Big|^k}{\frac{1}{x}}=0 \text{ for }k>1$$

Why is it true that $\exists x\in (0,b]$ such that $\Big|\ln\frac{1}{x}\Big|^k \leq \frac{1}{x}$ and $0<b<1$?