How to get value of $\lim\limits_{n \to \infty} \frac{\sqrt[n]{n!}}{n}$ without using Stirling-formula? The problem $$\lim\limits_{n \to \infty}\frac{\sqrt[n]{n!}}{n}$$ was a for-fun-only exercise given by our Calculus Professor. I was able to solve it quite easily with the use of the Stirling-formula, but can't figure out if it can be done in a different way, possibly by brute force spiced with recognizing the Euler-number in the proccess. I got to $$\lim\limits_{n \to \infty} \sqrt[n]{1 \cdot (1-1/n) \cdot (1-2/n) \cdot ... \cdot (1-(n-1)/n) }$$ by simply bringing the $n$ inside the n-th root sign, but I believe this is not the correct path.
I have no other ideas.

Start as
$$ \frac{(n!)^{1/n}}{n} = e^{ \frac{1}{n}\ln n! - \ln n } \sim e^{ \frac{n\ln n-n+1}{n}-\ln n }= e^{-1+1/n} \longrightarrow ...  $$
