I have thought along the same lines but I have been shown only frustrating non-results.
So that you may share my frustration inf * 1/(inf) is undefined:
consider limits as n tends to infinity:
if you have n one one side and 1/nē on the other, you consider the limit of their product which is simply the limit of 1/n ie 0 (or 1/infinity)
if you had nē on top and n on the bottom you would fnd the result to be infinity
if you had a*n on top and n on the bottom you would get a
Frustratingly, all of these examples correspond to inf/inf and have different answers.
To find practical solution you need to have a non infinite comparison between the two, usually just find an n and consider the limit of the product.
The interesting thing I did gain from this which is not fully relayed is the fact that some infinite spaces are larger than others, which is a handfull to visualise.