Dangerous Limit

Calculus Level 3

$\Large \displaystyle \xi(k) = \lim_{x\rightarrow \infty} \left( \dfrac{ (k x)!}{x^{kx}} \right)^{1/x} , \quad k \in \mathbb N$

Choose the correct option.

\xi(1) = \dfrac{1}{e}

\xi(2) =\dfrac{1}{e}

None of these

\\xi(1) = \dfrac{2}{e}

\xi(2) = \dfrac{4}{e}