Are all slow-varying functions slower than all polynomials?

Question

I'm wondering if all increasing slow varying functions are slower than all powers of x. It is easy to show that all positive powers of x are not slow varying but I'm wondering if it is possible to show that if for all positive real k and some arbitrary increasing function f $\lim_{x\to\infty}\frac{x^k}{f(x)}=\infty$ would imply $\lim_{x\to\infty}\frac{f(ax)}{f(x)}=1$.