Let $$f(x) = \left\{ {\matrix{ {{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr {0,} & {if\,x = 0} \cr } } \right.$$, $$(p,q \in R)$$. Then, Lagrange's mean value theorem is applicable to f(x) in closed interval [0, x]

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

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