If `f(x)=(tan(pi/4-x))/(cot2x)` for `x!=pi/4,` find the value which can be assigned to `f(x)` at `x=pi/4` so that the function `f(x)` becomes continuous every where in `[0,pi/2]dot`

Options

A. 1
B. `1//2`
C. 2
D. none of these

Answer

B

Explanation

For f(x) to be continous at `x=(pi)/(4)`, we must have `underset(x to pi//4)limf(x)=f((pi)/(4))`
`underset(x to pi//4)lim (tan(pi//4-x))/(cot 2x)=f((pi)/(4))`
`underset(x to pi//4)lim (tan(pi//4-x))/(tan (pi//2-2x))=f((pi)/(4))`
`Rightarrow (1)/(2) underset(x to pi//4)lim ({tan (pi//4-x)/((pi//4-x))})/({(tan2(pi//4-x))/(2(pi//4-x))})=f((pi)/(4)) Rightarrow (1)/(2)=f((pi)/(4))`