【例3】二元函数f(x,y)在点(0,0)处可微的一个充分条件是A. lim_(x to 0)[f(x,y)-f(0,0)]=0.B. lim_(x to 0)(f(x,0)-f(0,0))/(x)=0,且lim_(y to 0)(f(0,y)-f(0,0))/(y)=0.C. lim_(x to 0)(f(x,y)-f(0,0))/(sqrt(x^2)+y^(2))=0.D. lim_(x to 0)[f_(x)^prime(x,0)-f_(x)^prime(0,0)]=0,且lim_(y to 0)[f_(y)^prime(0,y)-f_(y)^prime(0,0)]=0.

A. $\lim_{x \to 0}[f(x,y)-f(0,0)]=0.$

B. $\lim_{x \to 0}\frac{f(x,0)-f(0,0)}{x}=0,$且$\lim_{y \to 0}\frac{f(0,y)-f(0,0)}{y}=0.$

C. $\lim_{x \to 0}\frac{f(x,y)-f(0,0)}{\sqrt{x^{2}+y^{2}}}=0.$

D. $\lim_{x \to 0}[f_{x}^{\prime}(x,0)-f_{x}^{\prime}(0,0)]=0,$且$\lim_{y \to 0}[f_{y}^{\prime}(0,y)-f_{y}^{\prime}(0,0)]=0.$