_ x 0 a^x-b^x c^x-d^x =

Step-by-step solution

The given limit is L = _ x 0 a^x - b^x c^x - d^x . Since the limit is of the form 0 0 as x 0 , we apply L'Hopital's rule by differentiating the numerator and the denominator with respect to x . The derivative of a^x is a^x a . Applying the rule: L = _ x 0 d dx (a^x - b^x) d dx (c^x - d^x) = _ x 0 a^x a - b^x b c^x c - d^x d . Evaluating the limit as x 0 : L = a^0 a - b^0 b c^0 c - d^0 d = a - b c - d . Using the property of logarithms x - y = ( x y ) : L = ( a b ) ( c d ) . This is equivalent to ( a b ) ( c d ) . A