$$\mathop {\lim }\limits_{x \to 0} {(\sin x)^{2\tan x}}$$ is equal to

Let for all x > 0, $$f(x) = \mathop {\lim }\limits_{n \to \infty } n({x^{1/n}} - 1)$$, then

If $$y = (1 + x)(1 + {x^2})(1 + {x^4})...(1 + {x^{2n}})$$, then the value of $$\left( {{{dy} \over {dx}}} \right)$$ at x = 0 is

$$\mathop {\lim }\limits_{x \to 1} {\left( {{{1 + x} \over {2 + x}}} \right)^{{{(1 - \sqrt x )} \over {(1 - x)}}}}$$ is equal to