Let f(x) be a function defined as f(x) = cases a e^ 2x - b (2x) - c x x^2 & if x 0 3 & if x = 0 cases If f(x) is continuous at x = 0 , then the value of a + b + c is:

Step-by-step solution

For f(x) to be continuous at x = 0 , we must have: _ x 0 f(x) = f(0) = 3 _ x 0 a e^ 2x - b (2x) - c x x^2 = 3 Using Maclaurin series expansions: e^ 2x = 1 + 2x + (2x)^2 2! + = 1 + 2x + 2x^2 + (2x) = 1 - (2x)^2 2! + = 1 - 2x^2 + Substituting these into the limit: _ x 0 a(1 + 2x + 2x^2) - b(1 - 2x^2) - c x x^2 = 3 Grouping the terms by powers of x : _ x 0 (a - b) + (2a - c)x + (2a + 2b)x^2 x^2 = 3 For the limit to exist and be finite, the coefficients of x^0 and x^1 in the numerator must be zero: a - b = 0 a = b 2a -