Given that for each $a \in (0,1)$,
$$\lim_{h \rightarrow 0^+} \int_{h}^{1-h} t^{-a}(1-t)^{a-1}\, dt$$
exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on $(0,1)$.
The value of $g\left(\frac{1}{2}\right)$ is\\
(A) $\pi$\\
(B) $2\pi$\\
(C) $\frac{\pi}{2}$\\
(D) $\frac{\pi}{4}$