If the function `f(x)={((1-cosx)/(x^(2))",","for "x ne 0),(k,"for " x =0):}` continuous at x = 0, then the value of k is

Options

A. 1
B. 0
C. `1/2`
D. -1

Answer

C

Explanation

Since ` lim _ ( x to 0 ) f ( x ) = f ( 0 ) `
` rArr lim _ ( x to 0 ) ( 1 - cos x ) /( x ^ 2) = k`
` rArr lim _ ( x to 0 ) ( - ( - sinx ) ) /( 2x ) = k`
[ using L' Hospital's rule ]
` rArr ( 1 ) /(2) lim _ ( xto 0) (sinx )/( x ) = k rArr k = ( 1)/( 2 ) `