Order in taking limits?

Question

So I am reading this physics paper that they define Kolmogorov entropy for dynamical systems as follows:

$$K=\lim\limits_{\epsilon\to 0}\lim\limits_{T\to \infty}\frac{I(\epsilon,T)}{T}$$

They emphasise on the face that the limit has been taken first on $T$. Mathematically speaking, I don't see why order in taking limits is meaningful. How can one even translate this kind ordering into formal definition of taking limits?

EDIT: Thanks to the replies, I realised its related to uniform convergence.

Answer 1

Hint: (From Rudin)

Try to evaluate $$\lim_{n \to \infty} \lim_{m \to \infty} \frac{m}{m+n}$$ and $$\lim_{m \to \infty} \lim_{n \to \infty} \frac{m}{m+n}$$