Limits Question 5
Question 5 - 2024 (29 Jan Shift 1)
$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^{2}} \int _{x^{3}}^{\left(\frac{\pi}{2}\right)^{3}} \cos \left(\frac{1}{t^{3}}\right) d t\right)$ is equal to
(1) $\frac{3 \pi}{8}$
(2) $\frac{3 \pi^{2}}{4}$
(3) $\frac{3 \pi^{2}}{8}$
(4) $\frac{3 \pi}{4}$
Show Answer
Answer (3)
Solution
Using L’hopital rule
$$ \begin{aligned} & =\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{0-\cos x \times 3 x^{2}}{2\left(x-\frac{\pi}{2}\right)} \\ & =\lim _{x \rightarrow \frac{\pi^{-}}{2}} \frac{\sin \left(x-\frac{\pi}{2}\right)}{2\left(x-\frac{\pi}{2}\right)} \times \frac{3 \pi^{2}}{4} \\ & =\frac{3 \pi^{2}}{8} \end{aligned} $$
