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

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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