Hard one, Huh?

$\displaystyle{\lim_{n\rightarrow \infty} \sqrt[n]{\prod_{k=0}^{n-1}\left( 1-\frac{k}{n} \right)}}$ $=\displaystyle{\exp{\left( \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}\ln \left( 1-\frac{k}{n} \right) \right)}}$

Then, by the definition of Riemann integral over a regular partition, the limit becomes

$\displaystyle{\exp\left( \int_0^1 \ln(1-x)\, \text{d}x \right)}$

$=\displaystyle{\frac{1}{e} }$

Answer is $exp(P)$

Where P is integration of $ln(1-x)$ from zero to 1 =-1