题目

求下列极限:
(I) $\mathop{\lim }\limits_{x \rightarrow \infty }\frac{{x}^{2} - x\sin x}{{x}^{2} + x\sin \frac{1}{x}}$;
(II) $\mathop{\lim }\limits_{x \rightarrow + \infty }{\left( \frac{{a}^{\frac{1}{x}} + {b}^{\frac{1}{x}} + {c}^{\frac{1}{x}}}{3}\right) }^{x}(a,b,c$ 为正数);
(III) $\mathop{\lim }\limits_{x \rightarrow 0}\frac{\ln \left( {{\sin }^{2}x + {\mathrm{e}}^{x}}\right) - x}{\ln \left( {{\mathrm{e}}^{2x} - {x}^{2}}\right) - {2x}}$;
(IV) $\mathop{\lim }\limits_{x \rightarrow 0}\frac{{\left( 1 + x\right) }^{\frac{3}{x}} - {\mathrm{e}}^{3}}{x}$ ;
(V) $\mathop{\lim }\limits_{x \rightarrow 0}\frac{{\mathrm{e}}^{\tan x} - {\mathrm{e}}^{x}}{{x}^{3}}$
(VI) $\mathop{\lim }\limits_{x \rightarrow 0}\cot x\left( {\frac{1}{\sin x} - \frac{1}{x}}\right)$
(VII) $\mathop{\lim }\limits_{x \rightarrow 0}{\left( 1 - {x}^{2}\right) }^{\frac{1}{1 - \sqrt{1 - {x}^{2}}}};$
(VIII) $\mathop{\lim }\limits_{{x \rightarrow {0}^{ + }}}{x}^{\sin x}$ .