`f(x) = {{:((1-coskx)/(x sinx), if x ne 0),(1/2, if x = 0):}` at `x = 0 `

Answer

N/a

Explanation

We have, `f(x) = {{:((1-coskx)/(x sinx), if x ne 0),(1/2, if x = 0):}` at `x = 0 `
At `x=0, LHL = underset(hrarr0^(-))(lim)(1-coskx)/(xsinx)=underset(hrarr0)lim(1-cos(0-h))/((0-h)sin(0-h))`

`= underset(hrarr0)(lim)(1-cos(-kh))/(-hsin(-h))`
`= underset(hrarr0)(lim)(1-coskh)/(hsinh) , [:' cos (-theta)= cos theta, sin (-theta)= - sin theta]`
`= underset(hrarr0)lim(1-1+2sin^(2)'(kh)/(2))/(hsinh), [:' cos theta = 1 - 2 sin^(2) ' (theta)/(2)]`
`= underset(hrarr0)(lim) (2sin^(2)'(kh)/(2))/(h sin h)`
`= underset(hrarr0)lim(2sin'(kh)/(2))/((kh)/(2)).(sin'(kh)/(2))/((kh)/(2)).(1)/((sinh)/(h)).(k^(2)h//4)/(h)`
`= (2k^(2))/(4) = (k^(2))/(2)` , [`:' underset(hrarr0)(lim)' (sinh)/(h) = 1]`
Also, `f(0) = 1/2 rArr (k^(2))/(2) = 1/2 rArr k +- 1`