If _ x 0 a x + b (1+x) + c x e^x x^2(e^x - 1) = 5 , where a, b, c R , then the value of a^2 + b^2 + c^2 is

Step-by-step solution

Given limit is _ x 0 a x + b (1+x) + c x e^x x^2(e^x - 1) = 5 Using Maclaurin series expansions: Denominator = x^2 ( x + x^2 2! + ) = x^3 + x^4 2 + Numerator = a ( x - x^3 3! + ) + b ( x - x^2 2 + x^3 3 - ) + cx ( 1 + x + x^2 2! + ) Numerator = (a + b + c)x + ( - b 2 + c )x^2 + ( - a 6 + b 3 + c 2 )x^3 + For the limit to exist and be finite, the coefficients of x and x^2 in the numerator must be zero, since the denominator's lowest degree term is x^3 . a + b + c = 0 ... (1) - b 2 + c = 0 b = 2c ... (2) The limit is