设 $f(x, y)$ 具有连续的偏导数，且
$$
f(t x, t y)=t^2 f(x, y), f(1,2)=0, f_x^{\prime}(1,2)=3
$$
求 $\lim _{x \rightarrow 0}\left[1+f\left(x-\sin x+1, \sqrt{1+x^3}+1\right)\right]^{\frac{1}{x-\sin x-1+\sqrt{1+x^3}}}$.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$