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

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Limits

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

equals

Answer: (B)

Rationalise the denominator by multiplying by

2 + 1 + cos x

, turning it into

2 - (1 + cos x) = 1 - cos x

. The limit becomes

x \to 0 lim \frac{sin ^{2} x ( 2 + 1 + cos x )}{1 - cos x}

, and since

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

the awkward factor cancels, leaving

x \to 0 lim (1 + cos x) (2 + 1 + cos x) = (2) (2 + 2) = 2 \cdot 22 = 42

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Single correct option

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