If _ x 0 (2x) + a (x) + b x x^5 is finite, then the value of (b - a) is equal to :

Step-by-step solution

The given limit is _ x 0 (2x) + a (x) + b x x^5 . Expanding the trigonometric functions using their Maclaurin series: (2x) = (2x) - (2x)^3 3! + (2x)^5 5! - = 2x - 4 3 x^3 + 4 15 x^5 - (x) = x - x^3 3! + x^5 5! - = x - 1 6 x^3 + 1 120 x^5 - Substituting these into the numerator: Numerator = (2x - 4 3 x^3 + 4 15 x^5 ) + a (x - 1 6 x^3 + 1 120 x^5 ) + bx Grouping the terms by powers of x : Numerator = (2 + a + b)x - ( 4 3 + a 6 )x^3 + ( 4 15 + a 120 )x^5 + For the limit to be finite when divided by x^5 , the coefficie