$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = $

The value of the limit $\mathop {\lim }\limits_{x \to 0} {\left\{ {{1^{\frac{1}{{{{\sin }^2}x}}}} + {2^{\frac{1}{{{{\sin }^2}x}}}} + \dots + {n^{\frac{1}{{{{\sin }^2}x}}}}} \right\}^{{{\sin }^2}x}}$ is

$\mathop {\lim }\limits_{n \to \infty } {\left[ {\frac{{\sin \left( n \right)}}{{{n^2}}} + \log \left( {\frac{{en + 1}}{{n + e}}} \right)} \right]^n}$ is equal to

$\lim _{x \rightarrow 0} \left( \frac{x \cdot 10^x - x}{1 - \cos x} \right)$ is equal to

Evaluate the limit: $\lim _{x \rightarrow 2} \frac{x^3-x^2-x-2}{2 x^3-3 x^2-3 x+2}$

Which of the following limits vanish? (where $[ \cdot ]$ denotes the greatest integer function)