First slide

Evaluation of definite integrals

Question

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

Moderate

A

1
B

0
C

$- 1$
D

2

Solution

Let $f (x) = \int_{0}^{x} \cos t^{2} d t$ and $g (x) = x$ . Then $f (0) = g (0) = 0$
$∴ \lim_{x \to 0} \frac{f (x)}{g (x)} = \lim_{x \to 0} \frac{f^{'} (x)}{g^{'} (x)}$

$\Rightarrow \lim_{x \to 0} \frac{\int_{0}^{x} \cos t^{2} d t}{x} = \lim_{x \to 0} \frac{\cos x^{2} .1 - \cos 0.0}{1} = \lim_{x \to 0} \frac{\cos x^{2}}{1} = \cos 0 = 1$

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