题目

(I) 当 $\left| x\right| < 1$ 时,求 $\mathop{\lim }\limits_{n \rightarrow \infty }\left( {1 + x}\right) \left( {1 + {x}^{2}}\right) \left( {1 + {x}^{4}}\right) \cdots \left( {1 + {x}^{{2}^{n}}}\right)$ ;
(II) 当 $\left| x\right| \neq 0$ 时,求 $\mathop{\lim }\limits_{n \rightarrow \infty }\cos \frac{x}{2}\cos \frac{x}{4}\cdots \cos \frac{x}{{2}^{n}}$ ;
(III) 求 $\mathop{\lim }\limits_{{x \rightarrow \frac{\pi }{2}}}\frac{\left( {1 - \sqrt{\sin x}}\right) \left( {1 - \sqrt[3]{\sin x}}\right) \cdots \left( {1 - \sqrt[n]{\sin x}}\right) }{{\left( 1 - \sin x\right) }^{n - 1}}$ .