The value of $\underset{\mathrm{x}\to 1}{lim} {\left(\tan \frac{\mathrm{\pi x}}{4}\right)}^{\tan \frac{\mathrm{\pi x}}{2}}$ is

$\underset{\mathrm{x}\to 0}{lim} {\left(\tan \frac{\mathrm{\pi x}}{4}\right)}^{\tan \frac{\mathrm{\pi x}}{2}}$

$$\begin{gathered} ={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \left(\tan \frac{\mathrm{\pi x}}{4}-1\right)\tan \frac{\mathrm{\pi x}}{2}} \\ ={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \left(\frac{\sin \frac{\mathrm{\pi x}}{4}-\cos \frac{\mathrm{\pi x}}{4}}{\cos \frac{\mathrm{\pi x}}{4}}\right)\frac{2\sin \frac{\mathrm{\pi x}}{4}\cos \frac{\mathrm{\pi x}}{4}}{\cos \frac{\mathrm{\pi x}}{2}}} \end{gathered}$$

$$\begin{gathered} ={\mathrm{e}}^{\frac{\underset{\mathrm{x}\to 1}{lim} 2{\sin }^2\frac{\mathrm{\pi x}}{4}-2\sin \frac{\mathrm{\pi x}}{4}\cos \frac{\mathrm{\pi x}}{4}}{\cos \frac{\mathrm{\pi x}}{2}}} \\ ={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \left(\frac{1-\cos \frac{\mathrm{\pi x}}{2}-\sin \frac{\mathrm{\pi x}}{2}}{\cos \frac{\mathrm{\pi x}}{2}}\right)} \end{gathered}$$

$$\begin{gathered} ={\mathrm{e}}^{\underset{\mathrm{x}\to 1}{lim} \frac{\sin \frac{\mathrm{\pi x}}{2}\times \frac{\mathrm{\pi }}{2}-\cos \frac{\mathrm{\pi x}}{2}\times \frac{\mathrm{\pi }}{2}}{-\sin \frac{\mathrm{\pi x}}{2}\times \frac{\mathrm{\pi }}{2}}} \\ ={\mathrm{e}}^{-1} \end{gathered}$$