First slide

Evaluation of definite integrals

Question

The value of $lim_{x \to 0} \frac{\int_{0}^{x} \sin t^{2} d t}{\sin x^{2}}$ is

Moderate

A

1
B

0
C

2
D

none of these

Solution

$\begin{aligned} lim_{x \to 0} \frac{\int_{0}^{x} \sin t^{2} d t}{\sin x^{2}} = lim_{x \to 0} \frac{\sin x^{2}}{\cos x^{2} \cdot 2 x} \\ = \frac{1}{2} lim_{x \to 0} \frac{\sin x^{2}}{x^{2}} \cdot x \cdot \cos x = 0 \end{aligned}$

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