Consider the function f(x,y)=2x^2+3y^2/x-y if x doesnt = y, and 0 if x=y.

Compute fpartialx(0,0) directly from the limit definition of a partial derivative: 

fpartialx(x0,y0)=lim as h->0 f(x0+h,y0)-f(x0,y0)/h