Example of functions where limits don't commute

Question

Can we have a function $f(h,k)$ such that $$\lim_{h\to 0}\lim_{k\to 0}f(h,k)\neq \lim_{k\to 0}\lim_{h\to 0}f(h,k)?$$

$f(h,,k) = \frac{h^{2} - k^{2}}{h^{2} + k^{2}}.$ – M. Strochyk Nov 12 '20 at 14:32 — M. Strochyk, Nov 12 '20 at 14:32

John Bentin · Accepted Answer · 2020-11-12T14:47:08.200

2

Say $$f(x,y)=\frac {x-y}{x+y}\quad(x+y\neq0).$$You can extend the domain to $\Bbb R^2$ by defining $f(x,y)=0$ when $x+y=0$.

edited Nov 12 '20 at 14:47

answered Nov 12 '20 at 14:32

John Bentin

score 1 · Answer 2 · edited Mar 24 '24 at 15:53

1

Moore-Osgood theorem deals with interchange iterated limits like this, See here and here

Simple counter example from the wiki:

$$ \lim_{x,y \to 0,0} f(x,y) = \lim_{x \to 0 } \lim_{ y \to 0} \frac{x^2}{x^2 +y^2}$$

More detailed discussion about this here

edited Mar 24 '24 at 15:53

Prem

answered Nov 12 '20 at 14:32

p.s: I have no idea about the technicalities of the theorem because my knowledge of it is from a time where I found a limit interchanging problem xD – tryst with freedom Nov 12 '20 at 14:34
@mithusengupta123: For some strong counterexamples and more refined theorems, see the two links given in my answer to Functions continuous in each variable (my 5th Stack Exchange answer; the July 2011 date had me curious, so I checked with my earliest answers). This non-commutativity of limits was an especially motivating issue for many significant developments in the late 1800s and early 1900s (e.g. the Baire category theorem, among other things). – Dave L. Renfro Nov 12 '20 at 15:04
Those papers are wayy beyond my current level but I have added that answer into mine :D – tryst with freedom Nov 12 '20 at 15:07
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wayy beyond my current level --- I thought as much (I'm older is all), but I was thinking others might be interested (or you or the OP in time). – Dave L. Renfro Nov 12 '20 at 15:09

2 Answers2