If $\beta = \lim_{x \rightarrow 0} \frac{e^{x^3} - (1 - x^3)^{1/3} + ((1 - x^2)^{1/2} - 1) \sin x}{x \sin^2 x}$,then the value of $6 \beta$ is

Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that $\lim _{x \rightarrow \infty} \frac{\sin(x^2)(\log_e x)^\alpha \sin(1/x^2)}{x^{\alpha \beta}(\log_e(1+x))^\beta} = 0$. Then which of the following is (are) correct?

If $f(x) = \begin{cases} \frac{\sin(1+[x])}{[x]}, & \text{for } [x] \neq 0 \\ 0, & \text{for } [x] = 0 \end{cases}$ where $[x]$ denotes the greatest integer function,then $\lim_{x \rightarrow 0^{-}} f(x)$ is equal to

$\mathop {\lim }\limits_{x \to \pi /2} \frac{{{a^{\cot x}} - {a^{\cos x}}}}{{\cot x - \cos x}} = $

The value of $\lim_{x \rightarrow 1} \frac{(x^{2}-1) \sin^{2}(\pi x)}{x^{4}-2x^{3}+2x-1}$ is equal to

The value of $\mathop {\lim }\limits_{x \to 0} \frac{{\log (1 + {x^3})}}{{{{\sin }^3}x}} = $