Question

$\lim_{x \to \frac{\pi}{2}} \left( \frac{\int_{x^3}^{(\pi/2)^3} (\sin(2t^{1/3}) + \cos(t^{1/3})) dt}{(x-\frac{\pi}{2})^2} \right)$

Answer 1

$\frac{9\pi^2}{8}$

Answer 2

The limit is of the indeterminate form $\frac{0}{0}$. L'Hopital's Rule is applied twice. The derivative of the numerator is found using the Fundamental Theorem of Calculus and the chain rule. After the first application, the limit remains $\frac{0}{0}$, necessitating a second application of L'Hopital's Rule. The derivatives of the resulting numerator and denominator are calculated, and evaluating them at $x = \frac{\pi}{2}$ yields the final answer.