下列等式中，正确的是
A. $\lim _{x \rightarrow 0}\left(1+\frac{\sin x}{x}\right)^{\frac{x}{\min x}}=1$
B. $\lim _{x \rightarrow \infty}\left(1-\frac{\sin x}{x}\right)^{-\frac{\operatorname{mix}}{x}}=1$
C. $\lim _{x \rightarrow \infty}\left(1+\frac{\sin x}{x}\right)^{-\frac{x}{\sin x}}=e$
D. $\lim _{x \rightarrow 0}\left(1+\frac{\sin x}{x}\right)^{\frac{\sin x}{x}}=e$