\displaystyle \lim_{(x, y) \to (0, 0)} \frac{f(x, y) - f(0, 0) - \begin{bmatrix} 1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}}{\| (x, y) \|} = \lim_{(x, y) \to (0, 0)} \frac{\frac{x^3 - y^3}{x^2 + y^2} - x + y}{\sqrt{x^2 + y^2}}