定义法

1. 判断 ${\displaystyle f_x^{\prime }(x_0,y_0),f_y^{\prime }(x_0,y_0)}$ 是否都存在？
2. 判断极限 ${\displaystyle \underset{\mathrm{\Delta }x\to 0,\mathrm{\Delta }y\to 0}{lim}\frac{[f(x_0+\mathrm{\Delta }x,y_0+\mathrm{\Delta }y)-f(x_0,y_0)]-[f^{\prime }(x_0,y_0)\mathrm{\Delta }x+f^{\prime }(x_0,y_0)\mathrm{\Delta }y]}{\sqrt[]{(\mathrm{\Delta }x{)}^2+(\mathrm{\Delta }y{)}^2}}}$ 是否为 0.