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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 $0$.

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user422489
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    The definitions are not equivalent. The second one is not a definition of big-O. What is your source for this? – Did Mar 05 '17 at 12:32
  • Page 3 of Asymptotic Notes. Also it sounds intuitive to me but cannot prove it. Can you explain why it is not equivalent – user422489 Mar 05 '17 at 12:39
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    These notes might be referring to the (true) fact that, if $\lim\limits_{x\to \infty} \frac{f(x)}{g(x)}$ exists and if it is finite then $f(x)=O(g(x))$. But this is not a characterization (hence, not a definition) of $O$-asymptotics since $f(x)=O(g(x))$ does not imply that $\lim\limits_{x\to \infty} \frac{f(x)}{g(x)}$ exists. – Did Mar 05 '17 at 12:46

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