Let \( \alpha \) be a positive real number. Let \( f:\mathbb{R}\rightarrow\mathbb{R} \) and \( g:(\alpha,\infty)\rightarrow\mathbb{R} \) be the functions defined by \( f(x)=\sin\!\left(\dfrac{\pi x}{12}\right) \) and \( g(x)=\dfrac{2\log_{e}(\sqrt{x}-\sqrt{\alpha})}{\log_{e}(e^{\sqrt{x}}-e^{\sqrt{\alpha}})} \). Then the value of \( \lim\limits_{x\to \alpha^{+}} f(g(x)) \) is __________.
