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 $n \rightarrow \infty$?

Let $n = m^2$. 
$a = \ln (\sqrt n)! = \ln 1 + \ln 2 + \cdots + \ln m$
$b = \ln \sqrt {n!} = \frac{1}{2}\{\ln 1 + \ln 2 + \cdots + \ln m + \ln (m+1) + \ln (m+2) + \cdots + \ln m^2\}$
$a - b = \frac{1}{2}\{\ln 1 + \ln 2 + \cdots + \ln m\} 
- \frac{1}{2} \{\ln (m+1) + \cdots + \ln (m+m) + \cdots + \ln m^2\}$ 
$a - b < \frac{1}{2}\{\ln 1 + \ln 2 + \cdots + \ln m\} 
- \frac{1}{2} \{\ln 1 + \ln 2 + \cdots + \ln m\} = 0$
Hence $\sqrt n! < \sqrt {n!}$.
