If L = _ x 0 a- a²-x² - x² 4 x⁴ , a>0 . If L is finite,then

Step-by-step solution

The given limit is L = _ x 0 a - a^2 - x^2 - x^2 4 x^4 . Using the binomial expansion for a^2 - x^2 = a 1 - x^2 a^2 = a ( 1 - x^2 2a^2 - x^4 8a^4 - x^6 16a^6 - ) for small x . Substituting this into the expression: L = _ x 0 a - a ( 1 - x^2 2a^2 - x^4 8a^4 - ) - x^2 4 x^4 L = _ x 0 a - a + x^2 2a + x^4 8a^3 + - x^2 4 x^4 L = _ x 0 x^2 ( 1 2a - 1 4 ) + x^4 8a^3 + O(x^6) x^4 For the limit L to be finite, the coefficient of x^2 must be zero, so 1 2a - 1 4 = 0 . Solving for a , we get 1 2a = 1 4 , which implies 2a = 4