If _ x 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to :

Step-by-step solution

Let x - 2 = t . As x 2 , t 0 . The given limit can be written as: _ t 0 ((t+2)^3 - 5(t+2)^2 + a(t+2) + b) ( t+1 - 1) _e(t+1) = m The denominator can be approximated for small t as: _ t 0 t+1 - 1 t _e(t+1) t t^2 = 1 2 1 t^2 = t^2 2 For the limit to be finite, the argument of the sine function in the numerator must have t^2 as its lowest degree term in its expansion. Let P(t) = (t+2)^3 - 5(t+2)^2 + a(t+2) + b P(t) = t^3 + 6t^2 + 12t + 8 - 5(t^2 + 4t + 4) + at + 2a + b P(t) = t^3 + t^2 + (a - 8)t + (2a + b - 12) For t