Definition of big $O$:
$f(n) = O(g(n))$ if there are positive constants $c$ and $k$, such that $0 ≤ f(n) ≤ cg(n)$ for all $n ≥ k$.
The limit definition of big $O$:
$f(n) = O(g(n))$ if $ \displaystyle\lim_{x\to \infty} \frac{f(n)}{g(n)} = c $ where $c \geq 0.$
I am having a hard time proving that these two definitions are equivalent. I think you can use the epsilon-delta definition of limit to prove these two definitions are the same but I'm not able to.
There is a related question which asks for Little o limit equivalence. But in the case of big $O$, the constant $c$ can be either $0$ or greater than $0$.