Let `f(x)=(tan(pi/4-x))/(cot2x),x!=pi/4` . The value which should be assigned to `f(x)` at `x=pi/4,` so that it is continuous everywhere is (a) `1` (b) `1/2` (c) 2 (d) none of these

Explanation

`f(x) = (tan(pi/4 - x))/(cot 2x)`

`f(pi/4) = (tan(pi/4 - pi/4))/(cot (pi/2))= 0/0`

`l = lim_(x-> pi/4) f(x) `

`l = lim_(x-> pi/4) (tan(pi/4 - x))/(cot 2 x) = 0/0`

`l = lim_(x-> pi/4) (sec^2(pi/4 - x)(-1))/(- cosec^2 (2x)*2) `

`l = lim_(x-> pi/4) (sec^2 ( pi/4 - x))/(2cosec^2 (2x))`

`l = (1)^2/(2(1)^2) = 1/2`

`l = 1/2 = f(pi/4)`

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