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JEE Main · 2014 · 06 Apr 2014 · Offline · Limits

lim_{x \to 0} \frac{s i n ( π c o s ^{2} x )}{x ^{2}}

is equal to

(A) $\frac{π}{2}$
(B)1
(C) $- π$
(D) $π$ ✓ Correct

Answer: (D)

Solution

Use

cos^{2} x = 1 - sin^{2} x

to write

sin (π cos^{2} x) = sin (π - π sin^{2} x) = sin (π sin^{2} x)

. Then

\frac{sin ( π sin ^{2} x )}{x ^{2}} = \frac{sin ( π sin ^{2} x )}{π sin ^{2} x} \cdot \frac{π sin ^{2} x}{x ^{2}}

. As

x \to 0

the first factor tends to

1

and

\frac{sin ^{2} x}{x ^{2}} \to 1

, so the limit is

π

.

JEE Main · 2014 · 06 Apr 2014 · Offline · Single correct option

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