$$ \lim\limits_{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3 $$

$\lim\limits_{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to :

Let $f, g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as

$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.$

$g(x)=\left\{\begin{array}{cc}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\ 1, & x=-1\end{array}\right.$

and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the

value of $\lim\limits_{x \rightarrow 1} g(h(x-1))$ is :

Let $$x=2$$ be a root of the equation $$x^2+px+q=0$$ and $$f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.$$

Then $$\mathop {\lim }\limits_{x \to 2{p^ + }} [f(x)]$$, where $$\left[ . \right]$$ denotes greatest integer function, is