设 $f(x, y)=\left\{\begin{array}{l}\frac{\sin \left(x^2+y^2\right)}{x^2+y^2} \arctan \left(1+x^2+y^2\right), x^2+y^2 \neq 0, \\ \frac{\pi}{2}, \quad x^2+y^2=0,\end{array}\right.$ 若平面区域 $D: x^2+y^2 \leqslant a^2$ ，则 $\lim _{a \rightarrow 0^{+}} \frac{\iint_D f(x, y) \mathrm{d} x \mathrm{~d} y}{\pi a^2}=$
A
$\frac{\pi}{2}$ ．
B
$\frac{\pi}{4}$ ．
C
$\frac{\pi}{8}$ ．
D
0 ．
E
F