calculus - Finding $lim_{xtoinfty} frac {(x!)^{frac 1 x}}{x}$

Find $\displaystyle \lim_{x\to\infty} \frac {(x!)^{\frac 1 x}}{x}$

I have no idea how to solve it, I can approximate it to be in $(0,1)$ by squeezing but getting to the solution $(\frac 1 e)$ seems like it would require a lot more. Is this an identity?

Note: no integrals nor gamma function.

Answer

Note this

$$\left( \frac{x!}{x^x} \right)^{1/x} = (a_x)^{1/x}$$

where $a_x = \frac{x!}{x^x}$ and then use the fact that

$$\lim_{x\to \infty} (a_x)^{1/x} = \lim_{x\to \infty} \frac{a_{x+1}}{a_x}$$

and the evaluation of limit will become easy

$$\lim_{x\to \infty} \frac{a_{x+1}}{a_x} = \lim_{x\to \infty} \frac{1}{(1+1/x)^x} = \frac{1}{e}.$$