Let A = _ x 0⁺ (1+ ^2 x )^ 1 2 x , then _ e A =

Step-by-step solution

Given the limit A = x 0⁺ (1 + ^2 x )^ 1 2x , which takes the indeterminate form 1^ . Apply the exponential transformation for limits of the form [f(x)]^ g(x) with f(x) 1 and g(x) : A = e^ x 0⁺ g(x) [f(x) - 1] . Substitute f(x) = 1 + ^2 x and g(x) = 1 2x to obtain: A = e^ x 0⁺ ^2 x 2x Evaluate the exponent L = x 0⁺ ^2 x 2x by letting y = x , so as x 0⁺ , y 0⁺ : L = 1 2 y 0⁺ ( y y )^2 = 1 2 (1)^2 = 1 2 Thus, A = e^ 1 2 . Compute _e A = _e (e^ 1 2 ) = 1 2 . Final answer: 1 2