Is the order of limits interchangeable?

[Jhenrique]

In the sense most ample and general of limits, the following identitie is true:
$$\\ \lim_A \lim_B = \lim_B \lim_A$$
?

 

[micromass]

No, it's not.

 

[jbunniii]

Simple counterexample: consider the function ##x : \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}## defined by
$$x(m,n) =
\begin{cases}
1 & \text{if }m > n \\
0 & \text{otherwise}
\end{cases}$$
For every ##m##, we have ##\lim_{n \rightarrow \infty}x(m,n) = 0## and therefore ##\lim_{m \rightarrow \infty}\lim_{n \rightarrow \infty}x(m,n) = 0##.

Similarly, for every ##n##, we have ##\lim_{m \rightarrow \infty}x(m,n) = 1##, and therefore ##\lim_{n \rightarrow \infty} \lim_{m \rightarrow \infty}x(m,n) = 1##.

 

[Jhenrique]

and exist general cases where ##\\ \lim_A \lim_B = \lim_B \lim_A## is true?

 

[micromass]

and exist general cases where ##\\ \lim_A \lim_B = \lim_B \lim_A## is true?

Yes, and that's actually what a giant part of real analysis is about: finding when you can switch two limits.

Please see Knapp's "Basic Real Analysis". In the first chapter he already gives ##2## general situations where it's true.
Aside from that, there are many specialized situations where it is also true, these are incredibly important theorems. A small selection:
http://en.wikipedia.org/wiki/Monotone_convergence_theorem#Lebesgue.27s_monotone_convergence_theorem
http://en.wikipedia.org/wiki/Dominated_convergence_theorem
http://en.wikipedia.org/wiki/Fubini's_theorem
http://en.wikipedia.org/wiki/Power_series#Differentiation_and_integration