I have two functions, $f(n)$ and $g(n)$, and I am trying to determine whether $f(n)$ is $O(g(n))$, $\Omega(g(n))$ or $\Theta(g(n))$. I am not sure about my answers. Help will be appreciated.
a) $f(n)=n + \log(\log^2n)$, $g(n)=100n + (\log(n))^2$. I believe here, $f(n)$ is $\Theta(g(n))$ because we can ignore the lower order terms in $f(n)$ and $g(n)$, and so we are left with $f(n)=n$, $g(n)=100n$, and so $f(n)=\Theta(g(n))$.
b) $f(n)=3n(\log(n!)) + n^2$, $g(n)= n^2\log(\log(n))$. I think here $f(n)$ is $O(g(n)$ because $g(n)$ dominates all the time because of the term $n^2\log(\log n)$.
Am i right in my answers? Thanks for the help.