_ x 0 e ^ x - e ^x x-x =

Step-by-step solution

Mean Value Theorem Application The limit evaluates as _ x 0 e^ x - e^x x - x . Consider f(t) = e^t , which is continuous and differentiable on R , with f'(t) = e^t . For x near 0 , apply the Mean Value Theorem on the interval between x and x . There exists c in (x, x) such that e^ x - e^x x - x = e^c . As x 0 , both x and x approach 0 , so c 0 . Therefore, the limit becomes _ c 0 e^c = e^0 = 1 . Taylor Series Verification Using expansions near x=0 : e^u = 1 + u + u^2 2 + u^3 6 + O(u^4) and x = x + x^3 3 + O(x^5) .