_ x 0 (7^x-1 )^4 ( x k ) (1+ x^2 3 ) 4 x =3( 7)^3

Step-by-step solution

Evaluate the limit L = _ x 0 (7^x - 1)^4 ( x k ) (1 + x^2 3 ) 4x given that it equals 3( 7)^3 . Apply standard limits _ x 0 7^x - 1 x = 7 , _ x 0 x x = 1 , _ x 0 (1 + x) x = 1 , and _ x 0 x x = 1 . Express each component in terms of these limits: Numerator: (7^x - 1)^4 = ( 7^x - 1 x )^4 x^4 ( 7)^4 x^4 . Denominator terms: ( x k ) = (x/k) x/k x k x k , (1 + x^2/3) = (1 + x^2/3) x^2/3 x^2 3 x^2 3 , 4x = 4x 4x 4x 4x . Substitute into L : L = _ x 0 ( 7)^4 x^4 (x/k)(x^2/3)(4x) = ( 7)^4 x^4 (4x^4)/(3k) = 3k ( 7)^4 4 . Se