Evaluate: $\cos \left[ \lim_{x \rightarrow \infty} \frac{2 \pi |x| + \pi x}{|x| - 3x} + \lim_{x \rightarrow 0} \frac{\cos \left( \frac{\pi}{2} \cos^2 x \right)}{x^2} \right]$

The value of $\lim_{x \rightarrow 0} \frac{|x|}{x}$ is

$\lim _{x \rightarrow \infty}\left(\frac{x+6}{x+1}\right)^{x+4}$ is equal to

If $\mathop {\lim }\limits_{x \to 0} \phi (x) = {a^3}, (a \ne 0)$; then $\mathop {\lim }\limits_{x \to 0} \phi \left( {\frac{x}{a}} \right)$ is equal to :-

If $[x]$ denotes the greatest integer $\leq x$,then $\lim_{n \rightarrow \infty} \frac{1}{n^3} \{[1^2 x] + [2^2 x] + [3^2 x] + \ldots + [n^2 x] \} = $

Consider the following statements:
Statement $1$: $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a} = 1$ (where $a+b+c \neq 0$).
Statement $2$: $\lim _{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2} = \frac{1}{4}$.