求极限 $\lim _{t \rightarrow 0^{+}} \lim _{x \rightarrow+\infty} \frac{\int_0^{\sqrt{t}} d u \int_{t^2}^t \sin y^2 d y}{\left[\left(\frac{2}{\pi} \arctan \frac{x}{t^2}\right)^x-1\right] \arctan t^{\frac{3}{2}}}$.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$
$\text{E.}$
$\text{F.}$