# Big Oh and Big Omega clarification

Can I get an explanation of:

Can g(n) be Big O of $n^{2}$ and also the Big O of $n^{3}$? (at the same time)

Can g(n) be Big Omega of $\Omega (n)$ and also be the Big O of $n$?

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Do you know the definition of Big O? –  Antonio Vargas Nov 5 '13 at 4:32

For your first question: if $g(n)=O(n^2)$, then $g(n)=O(n^{2+k})$ for all $k>0$. Indeed, $g(n)=O(n^2)$ means that $$\limsup_{n\to\infty}\frac{|g(n)|}{n^2}<\infty,$$ and we also have $$\frac{|g(n)|}{n^{2+k}}\leq\frac{|g(n)|}{n^2}$$ if $k>0$.
The answer to your second question is yes. After all, it is possible that $g(n)=n$.
Did you mean to put $n^{3}$ under the lim so that it's $n^{3}$ dividing $g(n)$ in order for the the limit to go on to infinity? What does the 'sup' mean? –  GivenPie Nov 5 '13 at 2:55
The limit doesn't have to exist, that's why one uses the limit superior in the defintion. Or. if you want, $g(n)$ being $O(n^2)$ means that there exist $K>0$ with $|g(n)|\leq K\,n^2$ for $n$ big enough; in that case, $n^3$ would also foot the bill. –  Martin Argerami Nov 5 '13 at 5:01