If the function f defined on `(pi/6,pi/3)` by `{{:((sqrt2 cos x -1)/(cot x -1)" , " x ne pi/4),(" is continuous,"),(" k , "x=pi/4 ):}`
then k is equal to

Options

A. `1/2`
B. 2
C. 1
D. `1/sqrt2`

Answer

A

Explanation

Given functions is
` f(x) = {{:((sqrt2 cos x -1)/(cot x -1)" ," x ne pi/4),(" k ,"x= pi/4):}`
`:' ` Function f(x) is continuous, so it is continuous at ` x = pi/4`.
`:." " f(pi/4) = underset( x to pi/4) lim f(x) `
` rArr" " k = underset( x to pi/4) lim (sqrt2 cos x -1)/( cot x -1) `
Put ` x = pi/4 + h," when " x to pi/4," then " h to 0`
` k = underset( h to 0) lim (sqrt2 cos (pi/4 + h)-1)/( cot (pi/4 + h) -1) `
` = underset( h to 0) lim (sqrt2[1/sqrt2 cos h - 1/sqrt2 sin h]-1)/((cot h -1)/(cot h + 1)-1)`
`[:' ` cos (x+y) = cos x cos y - sin x sin y and cot (x+y) = `(cot x cot y-1)/(cot y+ cot x) ]`
` = underset( h to 0) lim (cos h - sin h -1)/((-2)/(1+cot h))`
`= underset( h to 0) lim [((1-cos h)+sin h)/(2sin h) (sin h+ cos h)]`
` = underset( h to 0) lim [(2sin^(2).h/2+2 sin. h/2 cos. h/2)/(4 sin. h/2 cos. h/2)(sin h+ cos h)]`
` = underset( h to 0) lim [(sin.h/2+ cos.h/2)/(2 cos.h/2) xx (sin h+ cos h)] rArr k = 1/2`