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 $\lim _ { x \rightarrow \alpha ^ { + } } f ( g ( x ) )$ is $\_\_\_\_$.
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 $\lim _ { x \rightarrow \alpha ^ { + } } f ( g ( x ) )$ is $\_\_\_\_$.