_ x 0 1+ x-1- x x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}=$

A
-1
✓
1
C
2
D
-2

✓

Answer

Correct option: B.

1

(B)
Applying L-Hospital's rule, we get
$\lim _{x \rightarrow 0} \frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}$
$=\lim _{x \rightarrow 0}\left(\frac{\cos x}{2 \sqrt{1+\sin x}}+\frac{\cos x}{2 \sqrt{1-\sin x}}\right)$
$=\frac{1}{2}+\frac{1}{2}$
$=1$
Alternate method:
$\lim _{x \rightarrow 0}\left[\frac{1+\sin x-1+\sin x}{x(\sqrt{1+\sin x}+\sqrt{1-\sin x})}\right]$
$=\lim _{x \rightarrow 0} \frac{2 \sin x}{x(\sqrt{1+\sin x}+\sqrt{1-\sin x})}$
$=1$

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