If _ x 0 1 + a| x| - 1 - b| x| x^2 = -8 , where a and b are positive real numbers, then the value of a + b is

Step-by-step solution

Given limit is _ x 0 1 + a| x| - 1 - b| x| x^2 = -8 Using the binomial expansion for 1+t = 1 + t 2 - t^2 8 + where t = a| x| , we get: 1 + a| x| = 1 + a 2 | x| - a^2 8 | x|^2 + Substituting this into the numerator: _ x 0 (1 + a 2 | x| - a^2 8 ^2 x + ) - 1 - b| x| x^2 _ x 0 ( a 2 - b )| x| - a^2 8 ^2 x + x^2 For the limit to exist and be finite, the coefficient of the lower-degree non-differentiable term | x| must be zero. Therefore: a 2 - b = 0 a = 2b The limit then reduces to the next term: _ x 0 - a^2 8 ^2 x x^2