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

We have,
$$\begin{gathered} {\tan }^{-1}\frac{\mathrm{x}}{\mathrm{\pi }}<\frac{\mathrm{\pi }}{3} \\ \begin{array}{l}\Rightarrow \tan \left({\tan }^{-1}\frac{\mathrm{x}}{\mathrm{\pi }}\right)<\tan \frac{\mathrm{\pi }}{3}\\ \Rightarrow \frac{\mathrm{x}}{\mathrm{\pi }}<\sqrt{3}\Rightarrow \mathrm{x}<\sqrt{3}\mathrm{\pi }=5.44\end{array} \end{gathered}$$
$\therefore$The maximum value of x is 5.44