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Question: Evaluate :\(\lim _ { x \rightarrow 0 }\) \(\left( \frac{\sin x}{x} \right)^{\frac{\sin x}{x - \sin x...

Evaluate :lim⁡x→0\lim _ { x \rightarrow 0 } (sin⁡xx)sin⁡xx−sin⁡x\left( \frac{\sin x}{x} \right)^{\frac{\sin x}{x - \sin x}}.

A

e–1

B

e

C

e2

D

None of these

Answer

e–1

Explanation

Solution

Limit = lim⁡x→0\lim _ { x \rightarrow 0 } (sin⁡xx)sin⁡xx1−sin⁡xx\left( \frac { \sin x } { x } \right) ^ { \frac { \frac { \sin x } { x } } { 1 - \frac { \sin x } { x } } }

Put = t; then t → 1 when x → 0.

∴ limit = lim⁡t→1\lim _ { t \rightarrow 1 } , which is of the form 1∞

= =

= (Using L’ Hospital’s rule)

= e–1.