Let $f(n) = 3^\frac{n}{2}$ and $g(n) = n^\sqrt{n}$.
One can easily see that for large $n$, both functions go towards $\infty$. So, one could use limit definitions for big $O$-Notation to calculate $\lim_{x\to \infty} \left(\frac{f(n)}{g(n)}\right)$.
However,
$$\begin{equation}\begin{aligned}
\lim_{x\to \infty} \left(\frac{f(n)}{g(n)}\right) &= \lim_{x\to \infty} \left(\frac{3^\frac{n}{2}}{n^\sqrt{n}}\right) \\
&= \lim_{x\to \infty} \left(\frac{3^\frac{n}{2}}{1} * \frac{1}{n^\sqrt{n}}\right) \\
&= \lim_{x\to \infty} (\infty * 0) \\
&= \mathord{?}
\end{aligned}\end{equation}\label{eq2}$$
Maybe I am missing something trivial but I don't know how to continue from here on out. Perhaps there is another way to solve this problem? Only looking for tips and not solutions!
Best Answer
I think I would compare their log's. $\ln f(n)$ is linear. $\ln g(x) = \sqrt{n} \ln n << \sqrt{n}\sqrt{n} = n$. So $f$ is bigger than $g$ by a good bit.