Limits of O(1/x) and O(x) as x approaches 0

[Avichal]

I'm a bit confused with limits of big-O terms. What should be the answer for following:-
1) limit of O(1/x) as x->0. O(1) maybe but I'm not sure.
2) limit of O(x) as x-> 0. O(1) or 0?

 

[Einj]

The capital O symbol means that you quantity "behaves" like some other quantity in a certain limit. So, $O(1/x)$ behaves as $1/x$ and therefore goes to infinity as x approaches 0. For the same reason $O(x)\to 0$ as x approaches 0.