If _ x 0 (1+a^3 )+8 e^ 1 / x 1+ (1-b^3 ) e^ 1 / x =2 , then

Step-by-step solution

Let L = _ x 0 (1+a^3)+8 e^ 1 / x 1+(1-b^3) e^ 1 / x . Consider the limit as x 0⁺ . As x 0⁺ , 1/x , so e^ 1/x . Dividing the numerator and denominator by e^ 1/x , we get: L = _ x 0⁺ (1+a^3)e^ -1/x + 8 e^ -1/x + (1-b^3) = 0 + 8 0 + (1-b^3) = 8 1-b^3 . Given L = 2 , we have 8 1-b^3 = 2 , which implies 1-b^3 = 4 , so b^3 = -3 , or b = -3^ 1/3 . Now consider the limit as x 0⁻ . As x 0⁻ , 1/x - , so e^ 1/x 0 . L = _ x 0⁻ (1+a^3)+8 e^ 1 / x 1+(1-b^3) e^ 1 / x = (1+a^3) + 0 1 + 0 = 1+a^3 . Given L = 2 , we have 1+a^3 = 2 ,