Evaluate $\lim_{x\to\infty}(ax)!/x^x$ in terms of $a$

Question

This is my first post here, so I hope the question is formatted correctly. Anyway, like the title says, I was wondering if there exists any real (or complex) value of $a$ such that:

$$\lim_{x\to\infty} \frac{(ax)!}{x^x} =1$$

I have tried looking for a solution graphically, but I wasn't able to find anything useful.

Answer 1

Stirling gives $(ax)! \sim \sqrt{2\pi ax}\left(\frac{ax}{e}\right)^{ax}$, so the answer is no. For $a = 1$ it converges to $0$ and for $a > 1$ it diverges to $+\infty$.