Those last two notations are, formally, the same. To call a part negligible, we say it's asymptotically bounded above by a constant multiple of this expression,

Those last two notations are, formally, the same. To call a part negligible, we say it's asymptotically bounded above by a constant multiple of this expression, which obviously goes away as we approach the limit. The first one is a colloquial alternative definition that would probably be considered "wrong" in formal writing.