First slide

Introduction to limits

Question

The value of $lim_{x \to 0} \frac{\sqrt{1 - \cos x^{2}}}{1 - \cos x},$ is

Moderate

A

$1 / 2$
B

2
C

$\sqrt{2}$
D

none of these

Solution

$lim_{x \to 0} \frac{\sqrt{1 - \cos x^{2}}}{1 - \cos x} = lim_{x \to 0} \frac{\sqrt{(\frac{1 - \cos x^{2}}{x^{4}})}}{(\frac{1 - \cos x}{x^{2}})} = \frac{\sqrt{\frac{1}{2}}}{\frac{1}{2}} = \sqrt{2} [∵ lim_{α \to 0} \frac{1 - \cos α}{α^{2}} = \frac{1}{2}]$

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