Chapter 13.4, Problem 1QCE

Assume that $f\left(x,y\right)$ is differentiable at $\left(x_0,y_0\right)$ and let $\Delta f$ denote the change in f from its value at $\left(x_0,y_0\right)$ to its value at $\left(x_0+\Delta x,y_0+\Delta y\right).$

(a) $\Delta f\approx \text{\_\_\_\_\_\_}$

(b) The limit that guarantees the error in the approximation in part (a) is very small when both $\Delta x\text{and}\Delta y$ are close to 0 is $\_\_\_\_\_\_.$