I was looking at the double limits for two variable function in $\mathbb R$, and so I decided to partition the set of all such functions into two disjoint subsets as follows:
$$\mathbb F={\{f(x,y):(x,y) \in \mathbb R^2}\}=\mathbb F_i \cup \mathbb F_j \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,(0)$$
$$\mathbb F_i(f(X,Y))={\Biggl\{f(x,y) \in \mathbb F:\lim _{x\rightarrow X } \left( \lim _{y\rightarrow Y}f \left( x,y \right) \right) \neq \lim _{y\rightarrow Y} \left( \lim _{x \rightarrow X }f \left( x,y \right) \right)}\Biggr\} \quad\quad\quad\quad\quad\quad\quad(1)$$
$$\mathbb F_j(f(X,Y))={\Biggl\{f(x,y) \in \mathbb F:\lim _{x\rightarrow X } \left( \lim _{y\rightarrow Y}f \left( x,y \right) \right) = \lim _{y\rightarrow Y} \left( \lim _{x \rightarrow X }f \left( x,y \right) \right)}\Biggr\} \quad\quad\quad\quad\quad\quad\quad\,(2)$$
Here are some examples of subsets of (1) and (2) for ${\{X,Y}\}={\{0,\infty}\}$:
$$ \left\{ {\frac {1}{xy}},xy,{{\rm e}^{xy}},{ {\rm e}^{-xy}},\ln \left( xy \right) ,\tan \left( xy \right) \right\} \subset \mathbb F_i \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\quad\quad\quad\quad\quad\quad\quad\quad\quad(1e)$$
$$ \left\{ {x}^{y},{\frac {x}{y}},{\frac {y}{x}},\cos \left( {\frac {y}{ x}} \right) ,\sin \left( {\frac {y}{x}} \right) ,\tan \left( {\frac {x }{y}} \right) ,\tan \left( {\frac {y}{x}} \right) \right\} \subset \mathbb F_j\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\quad\quad\quad(2e)$$ And in observing the obvious trend that can be seen in these examples for many other functions, I made the following statements:
$$S_0:\lim _{x\rightarrow X} \left( \lim _{y\rightarrow Y}g \left( x,y \right) \right) =\lim _{y\rightarrow Y} \left( \lim _{x\rightarrow X }g \left( x,y \right) \right)$$
$$S_1: X=\lim _{y\rightarrow Y}\Bigr(\frac{1}{Y}\Bigl)\, \lor \, X=\lim _{x\rightarrow X}\Bigr(\frac{1}{Y}\Bigl) $$ $$S_2:\lim _{x\rightarrow X} \left( \lim _{y\rightarrow Y}(f(g \left( x,y )\right)) \right) =\lim _{y\rightarrow Y} \left( \lim _{x\rightarrow X }f(g \left( x,y \right)) \right) $$
$$S_0 \land S_1 \Rightarrow S_2$$
So my first question is:
1) Are the above assertions always the case, and what functions serve as counter examples?
And my second question is:
2) Does there exist any such case examples for which $\mathbb F_i \neq {\{\,}\}$ is true but $S_1$ is false?