I have a problem with proving that the limit as x goes to infinity for lnx/x is 0. Take the most basic approach:
Note that the derivative of lnx is 1/x whereas x has a derivative of 1. Hence, lnx is outgrown by x as x assumes larger and larger values (basically concavity). Hence, lnx/x will crash to 0. Or will it??
Clearly the function is asymptotic. But is this by itself proof? Have we truly established that it is asymptotic to precisely 0?
I know this may sound a bit ridiculous, but needs there not to also be a step establishing that it will also be asymptotic to 0?
Additionally, how would one carry out such a logical step?