1. The problem statement, all variables and given/known data I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my problem here is that if I just compute the derivative in the standard way and then take the limit (x,y)-->(0,0) I get that the derivatives don't exist at the origin. 2. Relevant equations df/dx=1+(2x^4+6x^2*y^2)/((x^2+y^2)^2 df/dy=-1-4x^3*y)/(x^2+y^2)^2 3. The attempt at a solution When I take the limit (x,y)->(0,0) and I put y=mx I get: df/dx=1+(2+6m^2)/(1+m^2)^2 df/dy=-1-(4m)/(1+m^2)^2 So, clearly both limits depend on the value of m, so they do not exist. So, why does the original method(using the definition of the derivative) wield the values 3 and -1 for each respectively? I also checked the answer to the problem and the answer gives 3 and -1 and quotes "by using the definition of the partial derivatives". Thanks in advance.