Answer
$\frac{\pi}{4}$
Work Step by Step
Let $t=\frac{x+1}{x-1}$
As $x$ approaches $\infty$, $t$ approaches the value 1. So the given limit becomes
$\lim\limits_{t \to 1} tan^{-1}(t)=\frac{\pi}{4}$
Hence
$\lim\limits_{x \to \infty} tan^{-1}(\frac{x+1}{x-1})=\frac{\pi}{4}$
