Let $f(x)=\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x}, x \neq \frac{\pi}{4}$. The value which should be assigned to f(x) at $x=\frac{\pi}{4}$, so that it is continuous there at, is

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For f(x) to be continuous at $x=\frac{\pi}{4}$, we must have

$\lim\limits_{x \rightarrow \pi / 4} f(x)=f\left(\frac{\pi}{4}\right)$

$\Rightarrow \lim\limits_{x \rightarrow \pi / 4} \frac{\tan (\pi / 4-x)}{\cot 2 x}=f\left(\frac{\pi}{4}\right)$

$\Rightarrow \lim\limits_{x \rightarrow \pi / 4} \frac{\tan (\pi / 4-x)}{\tan (\pi / 2-2 x)}=f\left(\frac{\pi}{4}\right)$

$\Rightarrow \frac{1}{2} \lim\limits_{x \rightarrow \pi / 4} \frac{\left\{\frac{\tan (\pi / 4-x)}{(\pi / 4-x)}\right\}}{\left\{\frac{\tan 2(\pi / 4-x)}{2(\pi / 4-x)}\right\}}=f\left(\frac{\pi}{4}\right) \Rightarrow \frac{1}{2}=f\left(\frac{\pi}{4}\right)$