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 $$\mathop {\lim }\limits_{x \to 0} g(x)$$ =

Let m and n be two positive integers greater than 1. If $$$\mathop {\lim }\limits_{\alpha \to 0} \left( {{{{e^{\cos \left( {{\alpha ^n}} \right)}} - e} \over {{\alpha ^m}}}} \right) = - \left( {{e \over 2}} \right)$$$ then the value of $${m \over n}$$ is _________.

The largest value of the non-negative integer a for which $$\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x - 1) + a} \over {x + \sin (x - 1) - 1}}} \right\}^{{{1 - x} \over {1 - \sqrt x }}}} = {1 \over 4}$$ is