If `f(x)={{:((1-sqrt2sinx)/(pi-4x)",",ifxne(pi)/(4)),(a",",if x=(pi)/(4)):}` in continuous at `(pi)/(4)`, then a is equal to :

Options

A. 4
B. 2
C. 1
D. `1//4`

Answer

D

Explanation

It is given that ` f (x) ` is continuous at ` x = (pi)/(4)`.
` therefore lim_(x to pi//4) f (x) = f ((pi)/(4))`
`rArr lim_(x to pi//4) (1 - sqrt2 sin x )/(pi - 4x) = a `
` rArr sqrt2 lim_(x to pi//4) ((1)/(sqrt2) - sin x )/( pi - 4x ) = a `
` rArr (sqrt2)/(4) lim_( x to pi//4) (sin ""(pi)/(4) - sin x)/( (pi)/(4) - x ) = a `
` rArr (1)/(4sqrt2) lim_(x to pi//4) (2 sin ((pi)/(8) - (x)/(2)) cos ((pi)/(8) + (x)/(2)))/(((pi)/(8) - (x)/(2))) = a `
` rArr (1)/( 2 sqrt2 ) cos ""(pi)/(4) = a rArr a = (1)/(4)`
ALITER ` lim_( x to pi//4) f (x) = f ((pi)/(4))`
` rArr lim_( x to pi//4) (1 - sqrt2 sin x )/( pi - 4 x ) = a `
` rArr lim_( x to pi//4) (-sqrt2 cos x )/(- 4) = a " " ` [ Using De L' Hospital' s Rule ]
` rArr (1)/( 2sqrt 2) cos""(pi)/(4) = a rArr a = (1)/(4)`.