Example of functions where limits don't commute

Question

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

Answer 1

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 $x+y=0$.

Answer 2

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