Limits, Continuity & Differentiability
Limit with Integral — L'Hôpital
nta_pyq_2024_apr
Grade 12
Question:
$\displaystyle\lim_{x\to\frac{\pi}{2}}\frac{\displaystyle\int_{x^3}^{(\pi/2)^3}(\sin(2t^{1/3})+\cos(t^{1/3}))\,dt}{\left(x-\dfrac{\pi}{2}\right)^2}$ is equal to
$\dfrac{5\pi^2}{9}$
$\dfrac{9\pi^2}{8}$
$\dfrac{11\pi^2}{10}$
$\dfrac{3\pi^2}{2}$
Step-by-Step Solution
Key Concept: Apply L'Hôpital (0/0 form). Differentiate numerator by Leibniz rule: $-3x^2(\sin(2x)+\cos x)$, denominator: $2(x-\pi/2)$. Still 0/0, apply L'Hôpital again.
Double L'Hôpital gives $\frac{9\pi^2}{8}$.
Correct Answer: 2