So far I learned:
$O(f(n)) := {g : Z≥0 → R : ∃c, n0 > 0∀n ≥ n0 : |g(n)| ≤ c · |f(n)|}$ $Ω(f(n)) := {g : Z≥0 → R : ∃c, n0 > 0∀n ≥ n0 : |f(n)| ≤ c · |g(n)|}$ $Θ(f(n)) := {g : Z≥0 → R : ∃c1,2, n0 > 0∀n ≥ n0 : c1 · |f(n)| ≤ |g(n)| ≤ c2 · |f(n)|}$
But on Wikipedia I found out that there are also other definitions of those notations (by limits).
E.g. $\limsup_{x \to a} \left|\frac{f(x)}{g(x)}\right| < \infty$ for $f \in \mathcal{O}(g)$. To be allowed using those definitions I need to prove them. But I can't find the proof online and I'm not able to prove it by myself. Do you know a way?