Landau Notation, Definitions: Limits vs. Corman's
To expand on Raphael's answer, both definitions are equivalent. The second definition is Landau-style (i.e. number theory style), while the first definition is computer science style. 

The Landau-style definition is clearly more succinct and I personally prefer it. There are two reasons to state the definition in the computer science style:


Textbook writers don't want to assume that their readers know calculus.
The Landau-style definition for $f = \Theta(g)$ is more awkward: $\lim\inf f/g &gt; 0$, $\lim\sup f/g &lt; \infty$.