\lim_{(h,k)\to (0,0)}{\frac{z(h,k)-z(0,0)-h\cdot \frac{\partial{z}}{\partial{x}}(0,0)-k\cdot \frac{\partial{z}}{\partial{y}}(0,0)}{\sqrt{h^2+k^2}}}=\lim_{(h,k)\to (0,0)}{\frac{\frac{h\cdot k}{h^2+k^2}+h+k-h-k}{\sqrt{h^2+k^2}}=\lim_{(h,k)\to (0,0)}{\frac{h\cdot k}{\sqrt{h^2+k^2} (h^2+k^2)}\neq 0