Can you solve it?

Calculus Level 4

$\large L = \lim_{x\to0} \left(\frac{(1+x)^{1/x}}{e}\right)^{\dfrac{1}{\sin{x}}}$

If $L = e^k$ , where $k$ is a real number, find the value of $k$ .

Clarification: $e$ denotes Euler's number, $e \approx 2.71828$ .