$\mathop {\lim }\limits_{n \to \infty } \sin (\pi \sqrt {{n^2} + 1} ) = $

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

For a certain value of $c$, $\mathop {Lim}\limits_{x \to - \infty } [(x^5 + 7x^4 + 2)^c - x]$ is finite and non-zero. The value of $c$ and the value of the limit are:

Evaluate the limit: $\lim _{x \rightarrow \infty}\left\{x-\sqrt[n]{\left(x-a_1\right)\left(x-a_2\right) \ldots\left(x-a_n\right)}\right\}$,where $a_1, a_2, \ldots, a_n$ are positive rational numbers.

Let $L = \lim_{x \rightarrow 0} \frac{a - \sqrt{a^2 - x^2} - \frac{x^2}{4}}{x^4}$,where $a > 0$. If $L$ is finite,then which of the following is true?

The value of $\lim _{x \rightarrow 0} \left( \frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}} \right)$ is equal to: