The value of _ x 0 ( a^ x +b^ x +c^ x 3 )^ 2 / x ,(a, b, c>0) is

Step-by-step solution

As x 0 , the base a^x+b^x+c^x 3 1 and exponent 2 x , giving 1^ indeterminate form. Using f(x)^ g(x) = e^ g(x)[f(x)-1] : L = e^ _ x 0 2 x ( a^x+b^x+c^x 3 -1 ) = e^ _ x 0 2 3 ( a^x+b^x+c^x-3 x ) = e^ _ x 0 2 3 ( a^x-1 x + b^x-1 x + c^x-1 x ) Using _ x 0 k^x - 1 x = k : L = e^ 2 3 ( a + b + c) = e^ 2 3 (abc) = e^ (abc)^ 2/3 = (abc)^ 2/3 This does not match options (1), (2), or (3). The answer is (4) None of these.