$\mathop {\lim }\limits_{x \to 0} \cos \frac{1}{x}$

If $S_1 = \sum_{r=1}^{n} r$,$S_2 = \sum_{r=1}^{n} r^2$,and $S_3 = \sum_{r=1}^{n} r^3$,then the value of $\lim_{n \rightarrow \infty} \frac{S_1(1 + \frac{S_3}{4})}{S_2^2}$ is

If $n$ is an integer,then $\mathop {\lim }\limits_{x \to n + 0} (x - [x]) = $

The value of $\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 - \cos {x^2}} }}{{1 - \cos x}}$ is

Let $A = \lim_{x \rightarrow 0^{+}} \left(1 + \tan^2 \sqrt{x}\right)^{\frac{1}{2x}}$,then $\log_{e} A = $

$\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {\frac{1}{2}(1 - \cos 2x)} }}{x} = $