Let P(x) be a polynomial of degree at most 3 such that _ x 0 e^x x - P(x) x^4 = L where L is a non-zero finite real number. Then the value of P(-3) is equal to :

Step-by-step solution

For the limit to exist and be a finite number, the Maclaurin series expansion of the numerator must have its first non-zero term of degree 4 or higher. This means that P(x) must exactly match the terms up to degree 3 in the expansion of e^x x . Expanding e^x and x using Maclaurin series: e^x = 1 + x + x^2 2! + x^3 3! + = 1 + x + x^2 2 + x^3 6 + x = 1 - x^2 2! + x^4 4! - = 1 - x^2 2 + Multiplying the two series and collecting terms up to degree 3 : e^x x = (1 + x + x^2 2 + x^3 6 ) (1 - x^2 2 ) + = 1 - x^2 2 + x - x^