Here, we consider the value of [math]f\left(x,y\right)[/math] as [math]\left(x,y\right)\to\left(0,0\right)[/math]. A common way to do this is to imagine the point [math]\left(x,y\right)[/math] approaching [math]\left(0,0\right)[/math] along a path. Often, the path is a straight line, but other paths are possible, too. As the point [math]\left(x,y\right)[/math] approaches [math]\left(0,0\right)[/math], we see what value [math]f\left(x,y\right)[/math] is approaching. If different paths cause [math]f\left(x,y\right)[/math] to approach different limits, then we know that the limit of of [math]f\left(x,y\right)[/math] as [math]\left(x,y\right)\to\left(0,0\right)[/math] does not exist.[br][br]In the figure, select an example and select a path. Then, click and drag the graph to rotate it and see the [math]z[/math]-values along the path.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]