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JEE Advanced · 2006 · 09 Apr 2006 · Limits XII

For

x > 0, lim_{x \to 0} ((sin x)^{1/ x} + (1/ x)^{s i n x})

is

(A)0
(B)-1
(C)1✓ Correct
(D)2

Answer: (C)

Solution

We evaluate the limit in two parts:

x \to 0 lim ((sin x)^{1/ x} + (1/ x)^{s i n x})

For the first term: as

x \to 0^{+}

,

sin x \to 0^{+}

and

1/ x \to + \infty

, so

(sin x)^{1/ x} \to 0

.

For the second term:

(1/ x)^{s i n x} = e^{s i n x l n (1/ x)} = e^{- s i n x l n x}

. As

x \to 0^{+}

,

sin x ln x \to 0

(using L'Hospital's rule on

lim_{x \to 0^{+}} sin x ln x

), so

e^{- s i n x l n x} \to e^{0} = 1

.

Therefore, the limit equals

0 + 1 = 1

.

JEE Advanced · 2006 · 09 Apr 2006 · Single correct option

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