Evaluate the limit: $\mathop {\lim }\limits_{x \to \infty } [x({a^{1/x}} - 1)]$,where $a > 1$.

If $\lim _{x \rightarrow 2} \frac{1+\sqrt{1+4 \log _2 x}}{2+\left(2 x+\sin ^2 x+2 \cos x\right)(2 x-4)}=m$,then $m(m-1)=$

The limiting value of the function $f(x) = \frac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x}$ as $x \to \frac{\pi}{4}$ is

$\lim \limits_{x}$ ${\rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to

If $\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{{a^{1/x}} + b}}{c}} \right)^x} = d$ (where $d$ is a non-zero finite value),then $(b + 1) \log_a d$ is equal to:

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.............