The background comes from information theory. There, one often sees a cost of communication that depends on the blocklength $n$ (how much information can be sent) and the error $\varepsilon$. Specifically, let some communication problem have a cost given by $C(n, \varepsilon)$. The asymptotic limit is usually defined when one first takes the blocklength $n\rightarrow\infty$ and then takes the error $\varepsilon\rightarrow 0$. Let it hold that
$$\lim_{\varepsilon\rightarrow 0}\lim_{n\rightarrow\infty}\frac{1}{n}C(n, \varepsilon) = C,$$
for some constant $C$. If one were to choose a sequence $\varepsilon_n$ where $\lim_{n\rightarrow\infty}\varepsilon_n = 0$, then does it also hold that
$$\lim_{n\rightarrow\infty}\frac{1}{n}C(n, \varepsilon_n) = C?$$
Or do there exist sequences $\varepsilon_n$ that would not satisfy the second equation?
EDIT: Added a missing $1/n$ in the second equation as per the comment.