Definition Of Limit For A Function Of Two Variables

The limit of a function of one variable was defined as follows.

if, given e > 0, there exists d > 0 such that |f(x) - L| < e whenever 0 < |x - c| < d.

More intuitively, it says that the function gets close to a particular number whenever the points at which it is being evaluated keep getting close to the particular point of interest.

Let us try to put that into a form for a function of two variables. Some of it is pretty obvious.

if, given e > 0, there exists d > 0 such that |f(x, y) - L| < e whenever 0 < something < d.

The only problem we have is what that "something" is. When we talk about points being "close" to each other, we are obviously talking about distance. In one dimension, the distance between two numbers a and b, is the absolute value of their difference, |a - b|. Notice we kept that one dimensional distance in our limit definition for functions of two variables when we said |f(x, y) - L| < e. This is okay since here we are talking about the distance between two numbers. Our problem is with two coordinate points getting close to a fixed two coordinate point. But we know the distance formula for the plane. Thus, our definition will be

if, given e > 0, there exists d > 0 such that |f(x, y) - L| < e whenever 0 < < d.

This helps us also see what our definition would look like for higher dimensions.

Unfortunately, there are times that these limit problems are much more difficult than limits of y = f(x). The reason is simple. With y = f(x), we can approach a point in the domain from the right or the left. With z = f(x, y), we can approach the point (a, b) along the line x = a, or the line y = b, or the line y = m(x - a) + b for any choice of m, or the parabola y = k(x - a)² + b for any choice of k, or along any number of other paths. And the limit must be the same on ALL of those paths.

Fortunately, most of the time that won't be a problem because we will be able to do things as we did back in Calculus 1. "Plug the numbers in and if you get something nice that is the answer."