If N = n! (n $$\in$$ N, n > 2), then find $$\mathop {\lim }\limits_{N \to \infty } \left[ {{{({{\log }_2}N)}^{ - 1}} + {{({{\log }_3}N)}^{ - 1}} + \,\,.....\,\, + {{({{\log }_n}N)}^{ - 1}}} \right]$$.

Use the formula $$\mathop {\lim }\limits_{x \to 0} {{{a^x} - 1} \over x} = {\log _e}a$$, to compute $$\mathop {\lim }\limits_{x \to 0} {{{2^x} - 1} \over {\sqrt {1 + x} - 1}}$$.

If f(a) = 2, f'(a) = 1, g(a) = $$-$$1 and g'(a) = 2, find the value of $$\mathop {\lim }\limits_{x \to a} {{g(x)f(a) - g(a)f(x)} \over {x - a}}$$.

Let R be the set of real numbers and f : R $$\to$$ R be such that for all x, y $$\in$$ R, $$|f(x) - f(y)| \le |x - y{|^3}$$. Prove that f is a constant function.