$\underset{x→0}{\lim}\left(\frac{(1+x)^{1/x}}{e}\right)^{\frac{1}{\sin x}}$ is equal to

$\frac{1}{\sqrt{e}}$

$\underset{x→0}{\lim}\left(\frac{(1+x)^{1/x}}{e}\right)^{\frac{1}{\sin x}}=e^{\underset{x→0}{\lim}\frac{(1+x)^{1/x}-e}{e\sin x}}$

$=e^{\underset{x→0}{\lim}\frac{(1+x)^{1/x}-e}{x}.\frac{x}{e\sin x}}=e^{\frac{-e}{2}.\frac{1}{e}}=e^{-\frac{1}{2}}=\frac{1}{\sqrt{e}}$