Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.

The limit definition of the partial derivatives $f_x\left(x_0,y_0,z_0\right)$, $f_y\left(x_0,y_0,z_0\right)$, and $f_z\left(x_0,y_0,z_0\right)$.

A function of three variables.

Let $f$be a function of three variables defined on an open set $S$containing the point $\left(x_0,y_0,z_0\right)$.

(a) The partial derivative with respect to $x$at $\left(x_0,y_0,z_0\right)$, denoted by $f_x\left(x_0,y_0,z_0\right)$, is the limit,

$\underset{h\to 0}{\lim }\frac{f\left(x_0+h,y_0,z_0\right)-f\left(x_0,y_0,z_0\right)}{h}$, provided that this limit exists.

(b) The partial derivative with respect to $y$at $\left(x_0,y_0,z_0\right)$, denoted by $f_y\left(x_0,y_0,z_0\right)$, is the limit,

$\underset{h\to 0}{\lim }\frac{f\left(x_0,y_0+h,z_0\right)-f\left(x_0,y_0,z_0\right)}{h}$, provided that this limit exists.

(c) The partial derivative with respect to $z$at $\left(x_0,y_0,z_0\right)$, denoted by $f_z\left(x_0,y_0,z_0\right)$, is the limit,

$\underset{h\to 0}{\lim }\frac{f\left(x_0,y_0,z_0+h\right)-f\left(x_0,y_0,z_0\right)}{h}$, provided that this limit exists.

Example:

A function, $f(x,y,z)=8-x^2-y^2-z^2$

$$\begin{gathered} f_x\left(2,1,1\right)=\underset{h\to 0}{\lim }\frac{f(2+h,1,1)-f(2,1,1)}{h} \\ =\underset{h\to 0}{\lim }\frac{\left(8-{\left(2+h\right)}^2-1^2-1^2\right)-\left(8-2^2-1^2-1^2\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{\left(8-4-h^2-4h-1-1\right)-\left(8-4-1-1\right)}{h} \\ =\underset{h\to 0}{\lim }\frac{-h^2-4h}{h} \\ =\underset{h\to 0}{\lim }-h-4 \\ =-0-4 \\ =-4 \end{gathered}$$