Limits

This is an experiment in using the World Wide Web to illustrate the intuitive notion of limits in functions of two variables.

If you have access to a browser that supports tables, you may prefer to view the Netscape version.


To say that z = f(x,y) has a limit L as (x,y) approaches (a,b) means that f(x,y) can be made as close as you wish to L by taking (x,y) sufficiently close to (a,b).

(We don't allow (x,y) = (a,b) when studying limits. Also, "as close as you wish" and "sufficiently close" can be given quite precise meanings).

For example, given any polynomial function such as f(x,y)=x - x^3 - xy^2 + x^3 y^2, for any point (a,b) the limit of f as (x,y) approaches (a,b) is simply f(a,b), just as for polynomials in one variable.

Note: We could even take out the point (a,b,f(a,a)) and the limit would still be f(a,b), as in the next example, where the point (0,0,f(0,0)) is missing.

The function f(x,y)= x^2 * y/(x^2 + y^2) isn't defined at (0,0), but it has a limit 0 as (x,y) approaches (0,0). Click on the picture of the function to see a QuickTime movie of the function spinning around (0,0). It's pretty clear that as you get close to (0,0), the function gets close to 0.

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However, consider the function f(x,y) = xy/(x^2 + y^2). As you get close to the origin the function value gets close to a whole range of numbers depending on what path you take.

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For example, as (x,y) approaches (0,0) along the line y = mx, a bit of algebra shows that the function approaches m/(1+m^2). For m=1 (that is, the line (x,x)) the function values are a constant 1/2; for m=-1, along the line (x,-x), the value is always -1/2. From this viewpoint you can better see how the graph of the function is made up of half-lines emanating from the z-axis.

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The upshot is that f(x,y) = xy/(x^2 + y^2) has no (single) limits as (x,y) approaches (0,0).

But Hold On, Pardner

It gets curiouser and curiouser for functions of several variables. Consider the function f(x,y) = x^2 * y/(x^4 + y^2). Along the line y=mx, f(x,mx) = mx/(x^2 + m^2) and as (x,y) gets close to (0,0), x must be getting close to zero. Along these linear paths, f(x,y) is approaching 0.

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However, as you know if you checked out the movie, this function lacks a single limit at (0,0)! Try following the parabola y=x^2. Then f(x,x^2) = x^4/(x^4 + x^4) = 1/2. Now any little disk around (0,0) will contain points of this parabola and also points of the line y=mx, where the function gets close to zero. Again, there's no single limit value.

Moral: The study of limits holds a lot more surprises for functions of two variables than for functions of a single variable.

E. Klotz and E. Magness
Jeremy Dilatush made the image for the last example, as well as the QuickTime movie software.
14 September 1995