Is $\lim_{n \to \infty} \lim_{l \to \infty} \ a_{n,l} = \lim_{l \to \infty} \lim_{n \to \infty} \ a_{n,l}\;$? Is $\lim_{n \to \infty} \lim_{l \to \infty} \ a_{n,l} = \lim_{l \to \infty} \lim_{n \to \infty} \ a_{n,l}\;$?
What happens if I replace limits with lim sups?
Thanks!

Brian already answered the first question very well, so I will answer the second.  If $\lim_{x\to\infty} x_{n}$ exists, then $$\limsup{x_{n}}=\liminf{x_{n}}=\lim_{x\to\infty}x_{n}.$$  Consider what would happen if this were not the case.  Then there would exist at least two different subsequential limits.   However, since every converging sequence has each subsequential limit converge to the same limit, clearly $\lim_{x\to\infty}x_{n}$ would not exist. 
So, if the limits do, in fact, exist, then the double $\limsup$s would be equivalent to the double limits.
