The value of _ x 1/ 2 [ 4 ⁻¹ x ]^ 1 1 - 2x^2 is equal to

Step-by-step solution

Let t = ⁻¹x , which implies x = t . As x 1/ 2 , t /4 . The given limit transforms to: _ t /4 ( 4t )^ 1 1 - 2 ^2 t Using the identity 2t = 1 - 2 ^2 t , the limit becomes: _ t /4 ( 4t )^ 1 2t As t /4 , the base approaches 1 and the exponent approaches . This is a 1^ indeterminate form. Using the standard result _ x a f(x)^ g(x) = e^ _ x a (f(x) - 1) g(x) , we get: L = e^K , where K = _ t /4 ( 4t - 1 ) 1 2t Let t = u + /4 . As t /4 , u 0 . K = _ u 0 4(u + /4) - 1 2(u + /4) K = _ u 0 4u + 1 - 1 (2u + /2) K = _ u 0 4u -