Limits and Continuity: Learn It 4

Functions of Three or More Variables

The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\,(x,\ y,\ z)[/latex] that gives the temperature at a physical location [latex](x,\ y,\ z)[/latex] in three dimensions. Or perhaps a function [latex]g\,(x,\ y,\ z,\ t)[/latex] can indicate air pressure at a location [latex](x,\ y,\ z)[/latex] at time [latex]t[/latex].

How can we take a limit at a point in [latex]\mathbb{R}^{3}[/latex]? What does it mean to be continuous at a point in four dimensions?

The answers to these questions rely on extending the concept of a [latex]\delta[/latex] disk into more than two dimensions. Then, the ideas of limits and continuity for functions of three or more variables follow naturally from the definitions for two variables.

[latex]\delta[/latex] ball in higher dimensions

Let [latex](x_0,\ y_0,\ z_0)[/latex] be a point in [latex]\mathbb{R}^{3}[/latex]. A [latex]\delta[/latex] ball in three dimensions consists of all points in [latex]\mathbb{R}^{3}[/latex] lying at a distance of less than [latex]\delta[/latex] from [latex](x_0,\ y_0,\ z_0)[/latex]:

[latex]\large{\big\{(x,\ y,\ z)\in\mathbb{R}^{3}\big|\sqrt{(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\,<\delta\big\}}[/latex].

 

For higher dimensions, add additional terms under the radical to correspond to each additional dimension.

For example, given a point [latex]P=(w_0,\ x_0,\ y_0,\ z_0)[/latex] in [latex]\mathbb{R}^{4}[/latex], a [latex]\delta[/latex] ball around [latex]P[/latex] can be described by

[latex]\large{\big\{(w,\ x,\ y,\ z)\in\mathbb{R}^{4}\big|\sqrt{(w-w_0)^{2}+(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\,<\delta\big\}}[/latex].

To show that a limit of a function of three variables exists at a point [latex](x_0,\ y_0,\ z_0)[/latex], we show that for any point in a [latex]\delta[/latex] ball centered at [latex](x_0,\ y_0,\ z_0)[/latex], the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.

Find [latex]\displaystyle\lim_{(x,\ y,\ z)\to(4,\ 1,\ -3)}\frac{x^{2}y-3z}{2x+5y-z}.[/latex]