A) 3
B) 2
C) 6
D) 1
Correct Answer: B
Solution :
Rationalize \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( x+2\sin x \right)\left( \sqrt{{{x}^{2}}+2\sin x+1}+\sqrt{{{\sin }^{2}}x-x+1} \right)}{{{x}^{2}}+2\sin x+1-{{\sin }^{2}}x+x-1}\]\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( x+2\sin x \right)\left( 2 \right)}{{{x}^{2}}+2\sin x-{{\sin }^{2}}x+x}\] \[\frac{0}{0}\]form using L' hospital \[\Rightarrow \underset{x\to 0}{\mathop{\lim }}\,\frac{\left( 1+2\cos x \right)\times 2}{2x+2\cos x-2\sin x\cos x+1}\]\[\Rightarrow \frac{2\times 3}{\left( 2+1 \right)}=2\]
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