If _ x 0 ( e^ ax - 1 x + bx )^ 2 x = e^5 , where a and b are real constants, then the value of a^2 + b^2 is :

Step-by-step solution

Let the given limit be L . For the limit to evaluate to a non-zero finite value e^5 as the exponent 2 x , the base must approach 1 . _ x 0 ( e^ ax - 1 x + bx ) = a + 0 = a Thus, a = 1 . The limit now takes the 1^ form. Using the standard formula for 1^ limits, L = e^K , where: K = _ x 0 2 x ( e^x - 1 x + bx - 1 ) K = _ x 0 2 x^2 ( e^x - 1 - x + bx^2 ) Using the Taylor series expansion for e^x = 1 + x + x^2 2! + K = _ x 0 2 x^2 ( 1 + x + x^2 2 - 1 - x + bx^2 ) K = _ x 0 2 x^2 ( ( 1 2 + b ) x^2 ) = 2 ( 1 2 + b ) = 1