In fact, by the ratio test, the limit of f(n)/f(n+1) must be greater than or equal to one.
The limit is >=1 if it exists, but it doesn’t have to exist. And the lim inf can be arbitrarily close to zero.
limit as n goes to infinity of a(n)/a(n+1) must be greater than or equal to 2
Ditto.
The limit is >=1 if it exists, but it doesn’t have to exist. And the lim inf can be arbitrarily close to zero.
Ditto.