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

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

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

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