$\mathop {\lim }\limits_{x \to \infty } \sqrt {\frac{{x + \sin x}}{{x - \cos x}}} = $

If $\alpha = \lim_{x \rightarrow 0^{+}} \left( \frac{e^{\sqrt{\tan x}} - e^{\sqrt{x}}}{\sqrt{\tan x} - \sqrt{x}} \right)$ and $\beta = \lim_{x \rightarrow 0} (1 + \sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $ax^2 + bx - \sqrt{e} = 0$,then $12 \log_e(a + b)$ is equal to.............

If $f(x) = \text{Sgn}(\text{Sgn}(\text{Sgn}(x)))$,then $\mathop {\lim }\limits_{x \to 0} f(x)$ is equal to :-

$\lim _{y \rightarrow 0} \frac{\sqrt{3+y^3}-\sqrt{3}}{y^3} = $

The value of the limit of $\frac{x^3 - 8}{x^2 - 4}$ as $x \to 2$ is

$\lim _{x \rightarrow 0} \frac{2x}{|x|+x^2} = $