If _ x 4 a - ( x + x)^8 b + 4x = L , where L is a finite non-zero real number, then the value of a + b + L is equal to

Step-by-step solution

For the limit to be finite and non-zero, the numerator and denominator must both approach 0 as x 4 . Setting the limit of the numerator to 0: a - ( 4 + 4 )^8 = 0 a - ( 2 )^8 = 0 a = 16 . Setting the limit of the denominator to 0: b + (4 4 ) = 0 b + = 0 b = 1 . Now, substitute a and b back into the limit: L = _ x 4 16 - ( x + x)^8 1 + 4x Let t = x + x . As x 4 , t 2 . Squaring both sides, t^2 = 1 + 2x 2x = t^2 - 1 . The denominator can be written as: 1 + 4x = 2 ^2 2x = 2(1 - ^2 2x) = 2(1 - (t^2 - 1)^2) = 2(2t^2 - t^