Prove the following

Question:

$\lim _{x \rightarrow 0} \frac{x+2 \sin x}{\sqrt{x^{2}-2 \sin x+1}-\sqrt{\sin ^{2} x-x+1}}$ is :

  1. 3

  2. 2

  3. 6

  4. 1


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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