General question about the existence (or not) of 2+ var functions

[iceblits]

I'm trying to figure out how to show that the limit of a function of two or more variables does not exist. I know that to do this we must show that the limit from 2 different pathways is not equal to the same thing, but, I want to know how to figure out what pathways to check, assuming you aren't using a graphing tool.
Similarly, how can we guarantee that all paths go to the same number and thereby show that the limit exists?

 

[SammyS]

It's not the sort of problem that's given to using one or two general techniques. That being said, there are a couple of techniques which can often be of help.

One is if the limit is being evaluated at the origin. Then writing the expression in polar coordinates, or spherical coordinates, or hyper-spherical coordinates can be helpful for both cases: convergence & divergence. If the limit is to be evaluated at some other point, do a coordinate translation to shift the point to the origin in the new coordinate system.

For easy cases you can try approaching the point at which the limit is being evaluated along a line written in some general form, e.g. in two dimensions approach the origin along the line, y = mx, in which the slope is the parameter, m. If the limit depends on m, then bingo, the limit does not exist & you're done. If the limit does not depend on m, maybe the limit exists.

Often this type of problem involves some rational expression. You may be able to set the expression to some constant value & solve for one of the variables. Try approaching the origin along a curve of the same or similar shape.

Nothing beats looking at a graph to get an idea about what's happening !

 

[iceblits]

Ok, thanks for the information :)