This is wrong; in the definition it should be "<= some constant", and it should ideally also mention that r and f need to positive, or else take absolute values

This is wrong; in the definition it should be "<= some constant", and it should ideally also mention that r and f need to positive, or else take absolute values of them. As an example of a bug in your definition, consider "sin(n) = O(1)". Also, if you say "r(n)/f(n) <= constant", then you don't need to take the limit as "n -> infty", and can talk about "all n".