Question

Evaluate the limit : \(\lim_{x \to \frac{\pi}{2}} \left( \frac{1}{\left( x - \frac{\pi}{2} \right)^3} \int_{\frac{\pi}{2}}^x \cos \left( \frac{1}{t^3} \right) \, dt \right)\)

• A. \(\frac {3\pi^2}{4}\)
• B. \(\frac {3\pi}{4}\)
• C. \(\frac {3\pi^2}{8}\)
• D. \(\frac {3\pi}{8}\)

Answer 1

The Correct Option is C

\(\text{Apply L'Hôpital's Rule: The given expression is:} \quad \lim_{x \to \frac{\pi}{2}} \frac{\int_{x}^{\frac{\pi}{2}} \cos \left( \frac{t}{2} \right) \, dt}{\left( x - \frac{\pi}{2} \right)^3}\)
Differentiate the Numerator and Denominator: Using the Fundamental Theorem of Calculus and L'Hôpital's Rule, we get:
\(= \lim_{x \to \frac{\pi}{2}} \frac{x^2 \cos \left( \frac{x}{2} \right)}{3 \left( x - \frac{\pi}{2} \right)^2}\)
Evaluate the Expression as \(x \to \frac{\pi}{2}\): As \(x \to \frac{\pi}{2}\), substitute appropriate values and simplify the expression:
\(= \frac{3\pi^2}{8}\)
So, the correct option is: \(\frac{3\pi^2}{8}\)