If `f(x)={((sin(cosx)-cosx)/((pi-2x)^2) ,, x!=pi/2),(k ,, x=pi/2):}` is continuous at `x=pi/2,` then `k` is equal to

Options

A. 0
B. `-(1)/(6)`
C. `-(1)/(24)`
D. `-(1)/(48)`

Answer

D

Explanation

It is given that f(x) is continous at `x=(pi)/(2)`
`therefore k=underset(x to pi//2)lim=f(x)`
`therefore k=underset(x to pi//2)lim=(sin(cos x)-cos x)/((pi-2x)^(3))`
`Rightarrow k=underset(x to pi//2)lim=(sin(cos x)-cos x)/((pi-2x)^(3))`
`Rightarrow k=underset(x to pi//2)lim=(sin(cos x)-cos x)/(cos^(3)x)xx(sin^(3)(pi//2)-x)/(8(pi//2-x)^(3))`
`Rightarrow k=(1)/(6)xx(1)/(8)=-(1)/(48)" "[therefore underset(x to 0)lim (sin x-x)/(x^(3))=-(1)/(6)]`