If `lim_(x to 1)((e^(k)-1)sin kx)/(x^(2))=4`, then k is equal to

Options

A. 2
B. `-2`
C. `pm2 `
D. `pm4`

Answer

C

Explanation

`lim_(x to oo ) ""((e^(kx)-1) sin kx)/( x^(2))=4 `
`implies lim_(x to 0)[(e^(kx) sin kx)/( x^(2))-(sin kx)/(x^(2))]=4 `
`implies lim_(x to 0)""[( e^(kx) cos kx * k +ke^(kx) sin kx)/(2 x)-( k cos kx )/(2 x)]=4 `
`implies lim_(x to 0)[( k^(2)e^(kx) cos kx-k^(2)e^(kx) sin kx +k^(2)e^(kx) sin kx +k^(2)e^(kx) cos ks )/(2) +(k^(2) sin kx )/( 2)]=4 `
`implies (k^(2)-0+0+k^(2))/(2)+0 =4 `
`implies k^(2)=4 `
`implies k= pm2`