The function `f(x)=(1-sinx+cosx)/(1+sinx+cosx)` is not defined at `x=pi`. The value of `f(pi)`, so that f(x) is continuous at `x=pi`, is

Options

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

Answer

C

Explanation

`lim_(x to pi) ((1+cosx)-sinx)/((1+cosx)+sinx)`
`=lim_(x to pi) (2cos^(2)x//2-2(sinx//2)cosx//2)/(2cos^(2)x//2+2(sinx//2) cosx//2)`
`=lim_(x to pi) (2"cos"^(2)(x)/(2)(1-"tan"(x)/(2)))/(2"cos"^(2)(x)/(2)(x+"tan"(x)/(2)))`
`=lim_(x to pi) tan((pi)/(4)-(x)/(2))=-1`
Since, f(x) is continuous at `x =pi`.
`therefore f(pi)=lim_(x to pi) f(x)=-1`