Indicating the complexity of functions

Question

I am not sure about my answer about the following question. Can anyone help?

I try to express whether $f(n)$ is $O(g(n))$, $\Omega(g(n))$ or $\Theta(g(n))$, where $f(n)=n^{0,1234}$ and $g(n)=n^{0,1233}$. I believe $f(n) = \Theta(g(n))$ but I am not so sure.

Thanks for the help.

Answer 1

Note that

$$\lim_{n\to\infty}\frac{f(n)}{g(n)}=\lim_{n\to\infty}\frac{n^{0,1234}}{n^{0,1233}}=\lim_{n\to\infty}n^{0,0001}=\infty\;,$$

So there cannot be a positive constant $c$ such that $f(n)\le cg(n)$ for all sufficiently large $n$. This shows that $f(n)$ is not $O\big(g(n)\big)$, which means that $f(n)$ cannot be $\Theta\big(g(n)\big)$, either. It also shows that $f(n)$ is $\Omega\big(g(n)\big)$; do you see why?