If _ x 0 e^ ax - (bx) - cx x^2 = 5 , where a and b are positive integers with a > b , then the value of a + b + c is equal to

Step-by-step solution

Given limit is _ x 0 e^ ax - (bx) - cx x^2 = 5 . Using the Maclaurin series expansions for e^ ax and (bx) : e^ ax = 1 + ax + a^2 x^2 2! + (bx) = 1 - b^2 x^2 2! + Substitute these into the numerator: e^ ax - (bx) - cx = (1 + ax + a^2 x^2 2 ) - (1 - b^2 x^2 2 ) - cx = (a - c)x + ( a^2 + b^2 2 )x^2 + For the limit to exist as x 0 , the coefficient of x in the numerator must be zero: a - c = 0 c = a The limit is then determined by the coefficient of x^2 : _ x 0 ( a^2 + b^2 2 )x^2 x^2 = a^2 + b^2 2 We are given that the