The limit _ x 0 24 ₀^ | x| ( t - 1) dt x^3

Step-by-step solution

Let L = _ x 0 24 ₀^ | x| ( t - 1) dt x^3 Using the Taylor series expansion for t , we have t - 1 - t^2 2 for small t . Evaluating the integral in the numerator: ₀^ | x| ( t - 1) dt ₀^ | x| (- t^2 2 ) dt = [ - t^3 6 ]₀^ | x| = - | x|^3 6 Substituting this back into the limit expression: L = _ x 0 24 ( - | x|^3 6 ) x^3 = _ x 0 -4 ( | x| x )^3 Now, we check the right-hand limit (RHL) and left-hand limit (LHL). For RHL ( x 0^+ ): x > 0 | x| = x RHL = _ x 0^+ -4 ( x x )^3 = -4(1)^3 = -4 For LHL ( x 0^- ): x LHL = _ x 0^