Let f(x) and g(x) be two functions defined on some subset of the real numbers. There are two definitions for $f \in o(g)\mbox{ as }x\to\infty\,$, according to Wikipedia:
- $\lim_{x \to \infty}\frac{f(x)}{g(x)}=0.$
- for every $M > 0$, there exists a constant $x_0$, such that $|f(x)| \le \; M |g(x)|\mbox{ for all }x>x_0. $
I was wondering if the two definitions are equivalent? Can it be possible that the second one is more general than the first one in that the limit of the ratio may not exist?
- Similar questions for $f \in \omega(g)$ and for $f \sim g$?
Thanks and regards!