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STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
Question
$\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}{x} = $
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Answer

b
(b) $\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}+\sqrt{1-\sin x}}$

Apply $L-$ Hospital‘s rule, 

$\mathop {\lim }\limits_{x \to 0} \frac{{\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}{x}$

$ = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{\cos x}}{{2\sqrt {1 + \sin x} }} + \frac{{\cos x}}{{2\sqrt {1 - \sin x} }} = \frac{1}{2} + \frac{1}{2} = 1.$

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