Let $f(x)=\left\{\begin{array}{rr}\frac{\tan x-\cot x}{x-\frac{\pi}{4},} & x \neq \frac{\pi}{4} \\ a, \quad & x=\frac{\pi}{4}\end{array}\right.$ The value of a so that f(x) is continuous at $x=\pi / 4$, is

4

If f(x) is continuous at $x=\frac{\pi}{4}$, then

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

$\Rightarrow a=\lim\limits_{x \rightarrow \pi / 4} \frac{\tan x-\cot x}{x-\frac{\pi}{4}}$

$\Rightarrow a=\lim\limits_{x \rightarrow \pi / 4} \frac{\sin ^2 x-\cos ^2 x}{\left(x-\frac{\pi}{4}\right) \sin x \cos x}$

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

$\Rightarrow a=2 \lim\limits_{x \rightarrow \pi / 4} \frac{\sin 2\left(\frac{\pi}{4}-x\right)}{2\left(\frac{\pi}{4}-x\right)} \times \frac{1}{\sin x \cos x}$

$\Rightarrow a=2 \times 1 \times 2=4$