问题
若有二元函数 $z$ $=$ $f(x, y)$, 且 $\phi$ $=$ $\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}$, 则，如何验证该二元函数的可微性？选项
[A]. $\lim _{\rho \rightarrow 0}$ $\frac{\Delta z-f_x^{\prime}(x, y) \Delta x+f_y^{\prime}(x, y) \Delta y}{\rho}$ $=$ $0$ $\Rightarrow$ $f(x, y)$ 可微[B]. $\lim _{\rho \rightarrow 0}$ $\frac{\Delta z-f_x^{\prime}(x, y) \Delta x-f_y^{\prime}(x, y) \Delta y}{\rho}$ $=$ $0$ $\Rightarrow$ $f(x, y)$ 可微
[C]. $\lim _{\rho \rightarrow 0}$ $\frac{\Delta z-f_x^{\prime}(x, y) \Delta x-f_y^{\prime}(x, y) \Delta y}{\rho^{2}}$ $=$ $0$ $\Rightarrow$ $f(x, y)$ 可微
[D]. $\lim _{\rho \rightarrow 0}$ $\frac{\Delta z+f_x^{\prime}(x, y) \Delta x-f_y^{\prime}(x, y) \Delta y}{\rho}$ $=$ $0$ $\Rightarrow$ $f(x, y)$ 可微
若 $\lim _{\rho \rightarrow 0}$ $\frac{\Delta z-f_x^{\prime}(x, y) \Delta x-f_y^{\prime}(x, y) \Delta y}{\rho}$ $=$ $0$,
或者：
$\lim _{\rho \rightarrow 0}$ $\frac{\Delta z – \frac{\partial z}{\partial x} \Delta x – \frac{\partial z}{\partial y} \Delta y}{\rho}$ $=$ $0$,
则函数 $z$ $=$ $f(x, y)$ 可微.