Explanation

(b)
Let $y = \lim_{x \rightarrow \infty}\left( \frac{\pi}{2} - \tan^{- 1}x \right)^{1/x}$
$\Rightarrow \ \ \ \ \ \ \ \log y = \lim_{x \rightarrow \infty}{\frac{1}{x}\log\left( \frac{\pi}{2} - \tan^{- 1}x \right)}$ $\left\lbrack \frac{\infty}{\infty}\ form \right\rbrack$
$= \lim_{x \rightarrow \infty}\frac{\left( - \frac{1}{1 + x^{2}} \right)}{\left( \frac{\pi}{2} - \tan^{- 1}x \right)}$ [using L’Hospital’s rule]
$= \lim_{x \rightarrow \infty}\frac{\frac{2x}{\left( 1 + x^{2} \right)^{2}}}{- \left( \frac{1}{1 + x^{2}} \right)} = \lim_{x \rightarrow \infty}\frac{- 2x}{1 + x^{2}}$
[using L’Hospital’s rule]
$\Rightarrow \ \ \ \ \log{y = 0}\ \ \ \ \ \Rightarrow \ \ \ \ \ \ \ \ y = 1$