Question

\( \lim_{x\to0} \frac{\int_0^{x^2} \sin(\sqrt{t}) \, dt}{x^2} \)

• A. \( \frac{2}{3} \)
• B. \( \frac{2}{9} \)
• C. \( \frac{1}{3} \)
• D. \( 0 \)
• E. \( \frac{1}{6} \)

Hint:

Use approximations like \( \sin x \approx x \) near 0.

Answer 1

The Correct Option is C

Concept: Small angle approximation.

Step 1: For small \( t \): \[ \sin(\sqrt{t}) \approx \sqrt{t} \]

Step 2: Approximate integral: \[ \int_0^{x^2} \sqrt{t} dt \]

Step 3: Integrate: \[ = \frac{2}{3} t^{3/2} \Big|_0^{x^2} \]

Step 4: Substitute: \[ = \frac{2}{3} (x^2)^{3/2} = \frac{2}{3} x^3 \]

Step 5: Divide: \[ \frac{\frac{2}{3}x^3}{x^2} = \frac{2}{3}x \to 0 \]