Which grows at a faster rate $\sqrt {n!}$ vs $(\sqrt {n})!$ when $n \rightarrow \infty$? Which grows at a faster rate $\sqrt {n!}$ vs $(\sqrt {n})!$ ? How to solve such type of questions considering $n \rightarrow \infty$?

As alluded to in the comments, $(\sqrt{n})!$ doesn't make sense, so I'm going to compare the growth of $n!$ to the growth of $\sqrt{(n^2)!}$. Or, equivalently, compare $(n!)^2$ to $(n^2)!$.
Let $a_n = \frac{(n!)^2}{(n^2)!}$. Then
\begin{align*}
\frac{a_{n+1}}{a_n} &= \frac{((n+1)!)^2 \div (n!)^2}{((n+1)^2)! \div(n^2)!} \\
&= \frac{(n+1)^2}{(n + 1)^2 ((n + 1)^2 - 1)((n + 1)^2 - 2) \ldots(n+1)} \\
&= \frac{1}{((n + 1)^2 - 1)((n + 1)^2 - 2) \ldots(n+1)}.
\end{align*}
So, the ratio between $(n!)^2$ and $(n^2)!$ very quickly approaches $0$, telling you that $(n^2)!$ grows much faster than $(n!)^2$.
