The function defined by f(x)= array cc x x + x & x>0 -5 k & x=0 4(1- 1-x ) x & x<0 array is continous at x=0, then k . equals

Step-by-step solution

For the function f(x) to be continuous at x = 0 , the left-hand limit, right-hand limit, and the value of the function at x = 0 must be equal. The right-hand limit is given by _ x 0⁺ f(x) = _ x 0⁺ ( x x + x ) = 1 + 1 = 2 . The left-hand limit is given by _ x 0⁻ f(x) = _ x 0⁻ 4(1 - 1 - x ) x . Multiplying the numerator and denominator by the conjugate (1 + 1 - x ) : _ x 0⁻ 4(1 - 1 - x )(1 + 1 - x ) x(1 + 1 - x ) = _ x 0⁻ 4(1 - (1 - x)) x(1 + 1 - x ) = _ x 0⁻ 4x x(1 + 1 - x ) = _ x 0⁻ 4 1 + 1 - x = 4 1 + 1 = 2 . Sinc