$\lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right)$

(C) equals $\frac{1}{\sqrt{2}}$

JEE Main · 2011 · 10 Apr 2011 · Offline · Limits

lim_{x \to 2} (\frac{1 - c o s { 2 ( x - 2 )}}{x - 2})

Answer: (D)

Use

1 - cos 2 θ = 2 sin^{2} θ

with

θ = x - 2

, so

1 - cos {2 (x - 2)} = 2 ∣ sin (x - 2) ∣

and the expression is

\frac{2 ∣ sin ( x - 2 ) ∣}{x - 2}

. As

x \to 2^{+}

x - 2 > 0

and the limit is

2

; as

x \to 2^{-}

x - 2 < 0

and the modulus flips the sign, giving

- 2

. The one-sided limits differ, so the limit does not exist.

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option

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