Question:
$\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^{2}-2 \sin x+1}-\sqrt{\sin ^{2} x-x+1}}$ is :
Correct Option: , 2
Solution:
Rationalize
$\lim _{x \rightarrow 0} \frac{(x+2 \sin x)\left(\sqrt{x^{2}+2 \sin x+1}+\sqrt{\sin ^{2} x-x+1}\right)}{x^{2}+2 \sin x+1-\sin ^{2} x+x-1}$
$\lim _{x \rightarrow 0} \frac{(x+2 \sin x)(2)}{x^{2}+2 \sin x-\sin ^{2} x+x}$
$\frac{0}{0}$ form using $L^{\prime}$ hospital
$\Rightarrow \lim _{x \rightarrow 0} \frac{(1+2 \cos x) \times 2}{2 x+2 \cos x-2 \sin x \cos x+1}$
$\Rightarrow \frac{2 \times 3}{(2+1)}=2$
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