The value of the limit

$$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{4\sqrt 2 (\sin 3x + \sin x)} \over {\left( {2\sin 2x\sin {{3x} \over 2} + \cos {{5x} \over 2}} \right) - \left( {\sqrt 2 + \sqrt 2 \cos 2x + \cos {{3x} \over 2}} \right)}}$$

is ...........

let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit

$$\mathop {\lim }\limits_{x \to {0^ + }} {{{{(1 - x)}^{1/x}} - {e^{ - 1}}} \over {{x^a}}}$$

is equal to a non-zero real number, is .............

The value of $${({({\log _2}9)^2})^{{1 \over {{{\log }_2}({{\log }_2}9)}}}} \times {(\sqrt 7 )^{{1 \over {{{\log }_4}7}}}}$$ is ....................

Let f : R $$ \to $$ R be a differentiable function such that f(0) = 0, $$f\left( {{\pi \over 2}} \right) = 3$$ and f'(0) = 1.

If $$g(x) = \int\limits_x^{\pi /2} {[f'(t)\text{cosec}\,t - \cot t\,\text{cosec}\,t\,f(t)]dt} $$

for $$x \in \left( {0,\,{\pi \over 2}} \right]$$, then $$\mathop {\lim }\limits_{x \to 0} g(x)$$ =