Entanglement measures and asymptotic continuity

I'm finishing my master studies in Theoretical Physics and my thesis is being about entanglement measures. I've read several papers in which all kind of properties that one would desire to have in a good entanglement measure are exhibit. One of these properties that one would consider fundamental is the asymptotic continuity of a entanglement measure. In particular, it's defined as follows:

"One entanglement measure E in d dimensions is asymptotic continuous if for two arbitrary states ρ and σ we've in the asymptotic regime n---->∞  that

||ρ - σ|| ---> 0

with

[E(ρ) - E( σ)]/log(d) ---> 0 "

I can understand that for two states which are close to each other, the value for the entanglement measure is expected to be similar, but what I don't understand is why we have to use the asymptotic regime. I've searched in various papers about entanglement measures but I haven't found a clear answer for the question.

Thank you everyone for reading me and I will be very grateful if you can help me with any contribution.