Limits Continuity And Differentiability

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



\(f(x)=\sin \left(\frac{\pi x}{12}\right) \quad \text { and } \quad g(x)=\frac{2 \log _{\mathrm{e}}(\sqrt{x}-\sqrt{\alpha})}{\log _{\mathrm{e}}\left(e^{\sqrt{x}}-e^{\sqrt{\alpha}}\right)} .\)



Then the value of \(\lim \limits_{x \rightarrow \alpha^{+}} f(g(x))\) is