`L e t` `f(x)=(1-tanx)/(4x-pi),x!=pi/4,x in [0,pi/2],` If`f(x)i s` continuous in `[0,pi/4],` then find the value of `f(pi/4)dot`

Options

A. 1
B. `(1)/(2)`
C. `-(1)/(2)`
D. `-1`

Answer

C

Explanation

It is given that f(x) is continous in `[0, pi//2]`. So, it is continous at `x=pi//4`.
`therefore f((pi)/(4))=underset(x to x+4)lim f(x)`
`Rightarrow f((pi)/(4))=underset(x to pi//4)lim (1-tanx)/(4x-pi)`
`Rightarrow f((pi)/(4))=underset(x to pi//4)lim (cos x-sin x)/(4cos x (x-pi//4))`
`Rightarrow f((pi)/(4))=-(sqrt2)/(4)=underset(x to pi//4)lim (((1)/(sqrt2)sin x-(1)/(sqrt2)cosx))/((x-(pi)/(4))cosx)`
`Rightarrow f((pi)/(4))=(1)/(2sqrt2) underset(x to pi//4)lim (sin(x-(pi)/(4)))/((x-(pi)/(4)))xx(1)/(cos x)`
`Rightarrow f((pi)/(4))=(1)/(2sqrt2) xx1xxsqrt2=-(1)/(2)`