设函数 $z=f(x, y)$ 连续, 且有 $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{f(x, y+1)-1-x-y}{\sqrt{x^2+y^2}}=0$, 求 $\lim _{n \rightarrow \infty}\left[f\left(0, \cos \frac{1}{n}\right)\right]^{n^2}$.
$\text{A.}$
$\text{B.}$
$\text{C.}$
$\text{D.}$