From the Prime Number Theorem, it follows that: $$\lim\limits_{n \rightarrow \infty}\sqrt[n]{n\#} = e$$
One of the standard definitions of $e$ as found here is that:
$$e = \lim\limits_{n \rightarrow \infty}\frac{n}{\sqrt[n]{n!}}$$
Based on this, would it be valid to conclude that:
$$\lim\limits_{n \rightarrow \infty}\frac{n}{\sqrt[n]{n!n\#}} = 1$$