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

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Limits

lim_{x \to 0} \frac{x + 2 s i n x}{x ^{2} + 2 s i n x + 1 - s i n ^{2} x - x + 1}

is:

Answer: (A)

Multiply top and bottom by the conjugate

x^{2} + 2 sin x + 1 + sin^{2} x - x + 1

. The denominator then rationalises to

(x^{2} + 2 sin x + 1) - (sin^{2} x - x + 1) = x^{2} + 2 sin x - sin^{2} x + x

. Near

x = 0

the conjugate sum tends to

2

, while both the numerator

x + 2 sin x

and the surviving part of the denominator behave like

3 x

. So the limit is

\frac{3 x \cdot 2}{3 x} = 2

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option

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