You need much more than limiting behavior to say anything about whether or not the processes are ‘similar’ in a useful way before that.
Perhaps the synthesis here is that while looking at asymptotic behaviour of a simpler system can be supremely useful, we should be surprised that it works so well. To rely on this technique in a new domain we should, every time, demonstrate that it actually works in practice.
Also, it’s interesting that many of these examples do have ‘pathological cases’ where the limit doesn’t match practice. And this isn’t necessarily restricted to toy domains or weird setups: for example, the most asymptotically efficient matrix multiplication algorithms are impractical (although in fairness that’s the most compelling example on that page).
Perhaps the synthesis here is that while looking at asymptotic behaviour of a simpler system can be supremely useful, we should be surprised that it works so well. To rely on this technique in a new domain we should, every time, demonstrate that it actually works in practice.
Also, it’s interesting that many of these examples do have ‘pathological cases’ where the limit doesn’t match practice. And this isn’t necessarily restricted to toy domains or weird setups: for example, the most asymptotically efficient matrix multiplication algorithms are impractical (although in fairness that’s the most compelling example on that page).