设 $f(t)=\iint_{x^2+y^2 \leqslant t^2} \arctan \left(1+x^2+y^2\right) \mathrm{d} x \mathrm{~d} y$, 则 $\lim _{t \rightarrow 0^{+}} \frac{f(t)}{\mathrm{e}^t-1-t}= $.
$\text{A.}$ $\frac{\pi}{2}$
$\text{B.}$ $\frac{\pi}{4}$
$\text{C.}$ $\frac{\pi^2}{2}$
$\text{D.}$ $\frac{\pi^2}{4}$