Limits


This is an experiment in using the World Wide Web to illustrate the intuitive notion of limits in functions of two variables. If your WWW browser doesn't seem to work with this (or loading is too slow), try the Non-Tabled 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 - x y^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,b)) 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 of 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. 329K
However, consider the function f(x,y) = xy/(x^2 + y^2). As you get close to the origin the function gets close to a whole range of numbers, depending on what path you take.

(You can learn how to make your own real-life model of this surface.)

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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. Take a look from the viewpoint of the xy-plane. From this viewpoint you can better see how the graph of the function is made up of half-lines emanating from the z-axis. 539K
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), so as (x,y) gets close to (0,0), x must be getting close to 0. Along linear paths to (0,0), f(x,y) is approaching 0.

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.

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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.

(Here's how to make your own model of the surface shown above.)


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.