Let a and b be positive integers such that _ x (ax^2 - 5x + 2) (bx+2)^ x 2 (3x^2 + 4x - 1) (bx-1)^ x 2 = 2 e ^ 3 4 The value of a + b is :

Step-by-step solution

The given limit can be split into the product of a rational function limit and an exponential function limit: _ x ax^2 - 5x + 2 3x^2 + 4x - 1 _ x ( bx+2 bx-1 )^ x 2 For the rational part, dividing the numerator and denominator by x^2 gives: _ x a - 5 x + 2 x^2 3 + 4 x - 1 x^2 = a 3 For the exponential part, we have a 1^ form: _ x ( 1 + 3 bx-1 )^ x 2 Using the standard limit _ x (1 + f(x))^ g(x) = e ^ _ x f(x)g(x) , we get: e ^ _ x ( 3 bx-1 x 2 ) = e ^ _ x 3x 2bx - 2 = e ^ 3 2b Multiplying the two parts, the total l