The value of `lim_(xrarr0) (sqrt(1-cosx^(2)))/(1-cos x)` is

Options

A. `(1)/(2)`
B. `2`
C. `sqrt2`
D. none of these

Answer

C

Explanation

`underset(xrarr0)(lim)(sqrt(1-cosx^(2)))/(1-cosx)`
`=underset(xrarr0)(lim)(sqrt(2sin^(2)(x^(2)//2)))/(2sin^(2)(x//2))`
`=(1)/(sqrt2)underset(xrarr0)(lim)(((sin(x^(2)//2))/(x^(2)//2))/(((sin(x//2))/(x//2))))xx(x^(2)//2)/(x^(2)//4)=sqrt2`