I think it is answered in http://math.stackexchange.com/questions/308144/o-notation-limsup-vs-lim (see the first answer). Indeed, `lim sup` appears correct. To answer myself:  the |g(x)/f(x)| doesn't have to have a limit, i.e. I guess the lim |g(x)/f(x)| might not exist, but it is bounded from above, so `lim sup` must exist and it is bounded. I think that's the idea. 

So I am ok with this change.