This question is related to: Landau Notation, Definitions: Limits vs. Cormen's.
Consider functions $f, \ g : N \rightarrow R^{\geq0}$.
For small-$o$, the definition: $$f(n)\in o(g(n)) \iff \forall c>0,\exists n_0>0, \forall n\geq n_0: f(n) < c \ g(n)$$
Can also be expressed as a limit (i guess this assumes the limit exists): $$f(n)\in o(g(n)) \iff \lim_{n\to\infty}\frac{f(n)}{g(n)}=0 $$
This can be verified introducing the definition for $\lim_{n\to\infty}$ on $N$, and then both definitions are identical.
Now, the "standard" definition for big-O would be something like (near exactly to Cormen's, except the codomain here is $R^{\geq0}$): $$f(n)\in O(g(n)) \iff \exists c>0,\exists n_0>0, \forall n\geq n_0: f(n) \leq c \ g(n) \qquad (1)$$
Im trying to see if the following is true (again, this assumes the limit exists): $$f(n)\in O(g(n)) \stackrel{?}{\iff} \exists c>0: \lim_{n\to\infty}\frac{f(n)}{g(n)}=c$$
Im not sure both directions of the equivalence are true. More specifically, i am having trouble with the $\Rightarrow$. According to this general answer, both directions should be true for $\Theta(g(n))$, wich is stronger than $O(\_)$. On the contrary, this answer (for roughly the same question, but in math SE) says big-$O$ cant be defined in that way (it was downvoted).
So, the questions are:
- How exactly def (1) implies the $\lim$ definition?:
$$\forall \epsilon>0,\exists n_0>0, \forall n\geq n_0: \left| \frac{f(n)}{g(n)}-c\right| < \epsilon$$
- Is the above still true considering generic functions $f, \ g : X \rightarrow R^{\geq0}$, where $X$ is some reasonable domain, maybe $Z$, $N^k$ or even $R^k$ (although reals are not turing-computable, im thinking in algorithms in the more general sense possible).
