Let α be a positive real number. Let f : R → R and g: (α, ∞) → R be the functions defined by
F(x) = sin\((\frac{\pi x}{12})\) and g(x) = \(\frac{2log_ e(\sqrt{x}-\sqrt{\alpha})}{log_e(e^{\sqrt{x}}-e^{\sqrt{\alpha}})}\)
Then the value of \(\lim\limits_{x \to \alpha^+}\) f(g(x)) is _______.
