So I just thought about the following problem, and I don't have the answer.
Consider a rectangle of lenght m, and width n, defined as such :
$$n(t)\times m(t) = 1, \forall t >0. $$
Consider that as t increases, n(t) decreases towards 0, meaning that m(t) must increase to preserve the constant area of the rectangle.
Does it makes sense to take the limit of this rectangle ? When $t \rightarrow +\infty, n(t) \rightarrow 0, m(t) \rightarrow + \infty.$ but at the same time, the area remains constant.
So
$$ \lim_{t \rightarrow +\infty} n(t)\times m(t) = 1$$
Intuitively, I'd say this rectangle converges towards a line, but how to deal with this non-null area ? What's wrong there ?
Nothing is wrong in this observation (it's actually a good one). What this shows is that one has to be very careful when taking limits (or interchanging two limits), and be very careful in determining which properties are "preserved under limits". In this particular example, you can interpret this as follows: define $f_n:\Bbb{R}^2\to \Bbb{R}$ to be $f_n:= \mathbf{1}_{[0,n]\times[0,\frac{1}{n}]}$ (where $\mathbf{1}_A$ is the function which is $1$ on $A$ and $0$ otherwise).
Then, for every $n$, we have $\int_{\Bbb{R}^2}f_n \, d \lambda = 1$ (i.e we're integrating with respect to Lebesgue measure on $\Bbb{R}^2$, which shows that each rectangle has area $1$). On the other hand, the pointwise limit of this sequence of functions is $\lim\limits_{n\to \infty} f_n = \mathbf{1}_{[0,\infty)\times \{0\}}$, and $\int_{\Bbb{R}^2}\mathbf{1}_{[0,\infty)\times \{0\}}\, d \lambda = 0$. In other words, \begin{align} \lim_{n\to \infty}\int_{\Bbb{R}^2}f_n \, d \lambda &= \lim_{n\to \infty}(1) = 1 \neq 0 = \int_{\Bbb{R}^2}\left(\lim_{n\to \infty}f_n\right)\, d \lambda \end{align} So, we can't just interchange limits with integration (which itself is a type of limit). In words, we can summarize this as "the limit of areas is not (necessarily) the area of the limit".
On a slightly tangential note, you should be careful to not switch limits arbitrarily. For instance, if you had a double sequence $\{a_{n,m}\}$, then in general $\lim\limits_{n\to \infty}\lim\limits_{m\to \infty}(a_{n,m}) \neq \lim\limits_{m\to \infty}\lim\limits_{n\to \infty}(a_{n,m})$. In fact, there are a bunch of theorems in analysis which deal exactly with the question of "when is it possible to interchange limits with other limits", such as whether it is possible to interchange limits and integration/ limits with differentiation and so on.