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 $f(x)=\sin \left(\frac{\pi x}{12}\right)$ and $g(x)=\frac{2 \log _e(\sqrt{x}-\sqrt{\alpha})}{\log _e\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)}$.
Then the value of $\lim _{x \rightarrow \alpha^{+}} f(g(x))$ is $\_\_\_\_$ .