If ${\tan }^{-1}\frac{x}{\pi }<\frac{\pi }{3}, x\in N$ then the maximum value of $x,$ is

We have, 

$\begin{array}{l} t{\mathrm{an}}^{-1}\frac{x}{3}<\frac{\pi }{3}\\ ={\tan }^{-1}\frac{x}{3}<{\tan }^{-1}\sqrt{3}\Rightarrow \frac{x}{3}<\sqrt{3}\Rightarrow x<3\sqrt{3}\end{array}$

Hence, the greatest integer value of $x$ is $5.$