Let L = _ x 0 ₀^x (e^ t^2 + a (2t) - b) dt x^5 . If L is finite, then the value of 10(a+b) + 60L is equal to :

Step-by-step solution

The given limit is L = _ x 0 ₀^x (e^ t^2 + a (2t) - b) dt x^5 . Since the limit is of the form 0 0 , we apply L'Hopital's rule and differentiate the numerator using the Leibniz rule: L = _ x 0 e^ x^2 + a (2x) - b 5x^4 Expanding the functions using their Maclaurin series: e^ x^2 = 1 + x^2 + (x^2)^2 2! + = 1 + x^2 + 1 2 x^4 + (2x) = 1 - (2x)^2 2! + (2x)^4 4! - = 1 - 2x^2 + 2 3 x^4 - Substituting these into the numerator: Numerator = (1 + x^2 + 1 2 x^4 ) + a (1 - 2x^2 + 2 3 x^4 ) - b Grouping the terms by powers of x