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Question: The value of \(\lim_{x \rightarrow 0}\left( \frac{\int_{0}^{x^{2}}{\sec^{2}tdt}}{x\sin x} \right)\) ...

The value of limx0(0x2sec2tdtxsinx)\lim_{x \rightarrow 0}\left( \frac{\int_{0}^{x^{2}}{\sec^{2}tdt}}{x\sin x} \right) is

A

3

B

2

C

1

D

0

Answer

1

Explanation

Solution

limx0ddx0x2sec2tdtddx(xsinx)=limx0sec2x2.2xsinx+xcosx\lim_{x \rightarrow 0}\frac{\frac{d}{dx}\int_{0}^{x^{2}}{\sec^{2}tdt}}{\frac{d}{dx}(x\sin x)} = \lim_{x \rightarrow 0}\frac{\sec^{2}x^{2}.2x}{\sin x + x\cos x}

(By L' –Hospital's rule)

= limx02sec2x2(sinxx+cosx)=2×11+1=1\lim_{x \rightarrow 0}\frac{2\sec^{2}x^{2}}{\left( \frac{\sin x}{x} + \cos x \right)} = \frac{2 \times 1}{1 + 1} = 1.