The value of _ x 4 _ x + x ^ 2 12 (t^5 - t^3) , dt 1 - ^3 2x is equal to

Step-by-step solution

The given limit is of the 0 0 form as x 4 . We apply L'Hospital's rule. Using the Leibniz integral rule, the derivative of the numerator with respect to x is: d dx ( _ x + x ^ 2 12 (t^5 - t^3) , dt ) = 0 - 12(( x + x)^5 - ( x + x)^3) ( x - x) The derivative of the denominator with respect to x is: d dx (1 - ^3 2x) = -3 ^2 2x (2 2x) Using the identity 2x = ^2 x - ^2 x = ( x - x)( x + x) , the derivative of the denominator becomes: -6 ^2 2x ( x - x)( x + x) Now, substitute these derivatives back into the limit: _ x 4