Question

The set of values of $x$ for which ${lim}_{x\to \infty }\frac{1}{1+{\left(\frac{4{\tan }^{-1}2x}{\pi }\right)}^{2n}}$ (where $n\in N$) is a non-zero finite quantity, is

• A. $[\frac{-1}{2},\frac{1}{2}]$
• B. $[0,\frac{1}{2}]$
• C. $(-1,1)$
• D. $[\frac{-1}{2},0]$

Answer 1

The correct option is A $[\frac{-1}{2},\frac{1}{2}]$

$-1\le \frac{4{\tan }^{-1}2x}{\pi }\le 1$
$\frac{-\pi }{4}\le {\tan }^{-1}2x\le \frac{\pi }{4}$
$\Rightarrow -1\le 2x\le 1orx\in [\frac{-1}{2},\frac{1}{2}]$