Section 14.6: Directional Derivatives & Gradient Vector

Show that if \(z = f(x, y)\) is differentiable at \(\mathbf{x}_0 = (x_0, y_0)\), then \(\lim_{\mathbf{x} \to \mathbf{x}_0} \frac{f(\mathbf{x}) - f(\mathbf{x}_0) - \nabla f(\mathbf{x}_0) \cdot (\mathbf{x} - \mathbf{x}_0)}{|\mathbf{x} - \mathbf{x}_0|} = 0\)

[Hint: Use Definition Directly]