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 of a function of two or three variables along a path.

A function of two or three variables.

(a) Let $f$be a function of two variables defined on an open set $S\subseteq {\mathrm{ℝ}}^2$. If the graph of the continuous vector function is a curve $C\subset S$such that then we define the limit of $f$as $\left(x,y\right)$approaches $\left(a,b\right)$along $C$, denoted by $\underset{\underset{C}{\left(x,y\right)\to \left(a,b\right)}}{\lim }f(x,y)$, to be $\underset{t\to t_0}{\lim }f\left(x\left(t\right),y\left(t\right)\right)$.

(b) Let $f$be a function of three variables defined on an open set $S\subseteq {\mathrm{ℝ}}^3$. If the graph of the continuous vector function is a curve $C\subset S$such that , then we define the limit of $f$as $\left(x,y,z\right)$approaches $\left(a,b,c\right)$along $C$, denoted by $\underset{\underset{C}{\left(x,y,z\right)\to \left(a,b,c\right)}}{\lim }f(x,y,z)$, to be $\underset{t\to t_0}{\lim }f\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)$.

The following schematic illustrates three such curves containing a point $\left(a,b\right)$in an open subset of ${\mathrm{ℝ}}^2$.