Question

The value of $\displaystyle\lim_{x \to0} \frac{\int\limits^{x^2}_{0} \sec^{2} tdt}{x \sin x} $

• A. 0
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (D)

$\displaystyle\lim_{x \to0} \frac{ \frac{d}{dx}\int\limits^{x^2}_{0} \sec^{2} tdt}{\frac{d}{dx}\left(x \sin x\right)} = \displaystyle\lim_{x \to0} \frac{\sec^{2} x^{2} .2x}{\sin x +x \cos x}$
(by L’ Hospital rule)
$ \displaystyle\lim_{x \to0 } \frac{2\sec^{2} x^{2}}{\left(\frac{\sin x}{x} + \cos x\right)} = \frac{2 \times1}{1+1} = 1$