I just have a general question on evaluating multivariable limits approaching the origin (0,0). If i determine that approaching from the x-axis and y-axis yields the same limit, my instructor said that it doesn't have to necessarily exist as it may not exist from another path.
So my question is, how do I go about finding such paths since there are literally an infinite number of them that approach the origin.
you can do it directly from the definition of the limit, but this is often "messy". but this amounts to the same thing as the polar coordinates version, really.
the idea is: that the magnitude of |f(x,y) - L| has to go to 0 as we consider the value of |f(x,y) - L| on "smaller and smaller" circles (well, disks, actually) surrounding the origin.
a common example is the following function:
. on a circle of radius r, .
it is not hard to show that on this circle, f takes on the values 1 and -1, (for example at (r,0) and (0,r)) so as we "shrink the circles", the value of f(x,y) does not converge.
contrast this with the behavior of . on a circle of radius r, , and as we shrink the circles (approaching (0,0) from "all directions at once") .
formally, let ε > 0, and choose δ = √ε.
then |(x,y) - (0,0)| < δ --> |(x,y)| < δ, that is:
, so
(since g is always non-negative).
given an epsilon, we can find a delta, so