My Calculus book defined a function $\displaystyle z=f(x,y)$ to be diffrenciable at point $\displaystyle (x_0,y_0)$. If and only if, the increment $\displaystyle f(x_0+\Delta x,y_0+\Delta y)$ can be expressed as:

$\displaystyle f(x_0+\Delta x,y_0+\Delta y)$=$\displaystyle f_x(x_0,y_0)\Delta x+f_y(x_0,y_0)\Delta y$$\displaystyle +\epsilon_1 \Delta x+\epsilon_2 \Delta y$

as $\displaystyle \Delta x\rightarrow 0, \Delta y\rightarrow 0$ then $\displaystyle \epsilon_1\rightarrow 0,\epsilon_2\rightarrow 0$.

My problem is that I understand it but I was hoping for a more rigorous definition. Can someone give one in terms of limits?