$\frac{x}{x-1}\xrightarrow{x\to {{1}^{-}}}\Rightarrow \frac{x}{x-1}\to -\infty $  $\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{\left| \tan \pi x \right|}{{{x}^{2}}-1}=\frac{0}{0}\xrightarrow{Taiin\,Alamat}\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{-\tan \pi x}{{{x}^{2}}-1}=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{-\tan (\pi x-\pi )}{(x-1)(x+1)}=\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,\frac{-\pi (x-1)}{(x-1)(x+1)}=-\frac{\pi }{2}$