If _ x 0 e^ x - e^x + a x^3 x^4 = c , where c is a finite real number, then the ordered pair (a, c) is equal to

Step-by-step solution

Using Maclaurin series expansions: x = x - x^3 6 + e^ x = 1 + ( x) + ( x)^2 2! + ( x)^3 3! + ( x)^4 4! + = 1 + (x - x^3 6 ) + 1 2 (x - x^3 6 )^2 + 1 6 (x - x^3 6 )^3 + 1 24 (x - x^3 6 )^4 + = 1 + x - x^3 6 + 1 2 (x^2 - x^4 3 ) + 1 6 (x^3) + 1 24 (x^4) + = 1 + x + x^2 2 - x^4 6 + x^4 24 + = 1 + x + x^2 2 - x^4 8 + Also, e^x = 1 + x + x^2 2 + x^3 6 + x^4 24 + Substituting these into the limit expression: _ x 0 (1 + x + x^2 2 - x^4 8 ) - (1 + x + x^2 2 + x^3 6 + x^4 24 ) + a x^3 x^4 = _ x 0 (a - 1 6 )x^3 - ( 1 8 + 1 2