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 $f (0) = g (0) = 0 .$
$\begin{array}{l} ∴ lim_{x \to 0} \frac{f (x)}{g (x)} = lim_{x \to 0} \frac{f^{'} (x)}{g^{'} (x)} \\ \Rightarrow lim_{x \to 0} \frac{\int_{0}^{x} \cos t^{2} d t}{x} = lim_{x \to 0} \frac{\cos x^{2} \cdot 1 - \cos 0 \cdot 0}{1} \\ = lim_{x \to 0} \frac{\cos x^{2}}{1} = \cos 0 = 1 . \end{array}$

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