The value of $\mathop {\lim }\limits_{x \to \infty } \frac{(x + 1)(3x + 4)}{x^2(x - 8)}$ is equal to

$\mathop {\lim }\limits_{n \to \infty } \cos \left( {\pi \sqrt {{n^2} + n} } \right)$,where $n \in I$,is

The value of $\lim _{x \rightarrow 0} \frac{x}{|x|+x^2}$ is .

Let $[.]$ denote the greatest integer function. Assertion $(A) : \lim_{x \rightarrow \infty} \frac{[x]}{x} = 1$. Reason $(R) : f(x) = x - 1, g(x) = [x], h(x) = x$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = \lim_{x \rightarrow \infty} \frac{h(x)}{x} = 1$.

The value of $\mathop {\lim }\limits_{x \to 0} \left( \left[ \frac{100x}{\sin x} \right] + \left[ \frac{99 \sin x}{x} \right] \right)$,where $[.]$ denotes the greatest integer function,is

If $\mathop {\lim }\limits_{x \to 5} \frac{{{x^k} - {5^k}}}{{x - 5}} = 500$,then the positive integral value of $k$ is