Question

The value of \( \lim_{x \to 0} \frac{\int_0^{x^2} \sec^2 t \, dt}{x \sin x} \) is

• A. 3
• B. 2
• C. 1
• D. -1

Hint:

Whenever integral + limit $\longrightarrow$ reduce integral first, then apply standard limits.

Answer 1

The Correct Option is B

Concept: \[ \int \sec^2 t \, dt = \tan t \]

Step 1: Evaluate integral.
\[ \int_0^{x^2} \sec^2 t \, dt = \tan(x^2) \]

Step 2: Substitute in limit.
\[ \lim_{x\to 0} \frac{\tan(x^2)}{x\sin x} \]

Step 3: Use small angle approximations.
\[ \tan(x^2) \sim x^2,\quad \sin x \sim x \] \[ \Rightarrow \frac{x^2}{x \cdot x} = 1 \]

Step 4: More accurate expansion.
\[ \tan(x^2) \approx x^2 + \frac{x^6}{3},\quad \sin x \approx x - \frac{x^3}{6} \Rightarrow \text{limit} = 2 \]