$\lim_{x \to 0} \frac{x + 2 \sin x}{\sqrt{x^2 + 2 \sin x + 1} - \sqrt{\sin^2 x - x + 1}}$
Step-by-Step Solution
Key Concept: Rationalise the denominator by multiplying by the conjugate. The denominator product $(x^2 + 2 \sin x + 1) - (\sin^2 x - x + 1) = x^2 + 2 \sin x - \sin^2 x + x$. Near $x = 0$: numerator $\to 0 + 0 = 0$, denominator $\to 0$. After rationalising, factor and substitute $x = 0$ in the non-vanishing parts to get $6$.
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Correct Answer: (4)