If _ x 0 (4^x - 1)^3 ( x 4 ) (1 + x^2 3 ) = 96( a)^b , then (a + b) =

Step-by-step solution

The given limit is _ x 0 (4^x - 1)^3 ( x 4 ) (1 + x^2 3 ) . Dividing the numerator and the denominator by x^3 , we get: _ x 0 ( 4^x - 1 x )^3 ( x 4 ) x (1 + x^2 3 ) x^2 Using the standard limits _ x 0 a^x - 1 x = a , _ x 0 x x = 1 , and _ x 0 (1 + x) x = 1 , we can rewrite the limit as: _ x 0 ( 4^x - 1 x )^3 ( x 4 ) x 4 1 4 (1 + x^2 3 ) x^2 3 1 3 Evaluating the limits, we get: ( 4)^3 1 1 4 1 1 3 = 12( 4)^3 Since 4 = (2^2) = 2 2 , we have: 12(2 2)^3 = 12 8( 2)^3 = 96( 2)^3 Comparing this with the given expression 96