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Question: \(\operatorname { Lim } _ { x \rightarrow 0 ^ { + } }\) \(\log_{\sin(x/2)}{}\) sin x is equal to –...

Lim⁡x→0+\operatorname { Lim } _ { x \rightarrow 0 ^ { + } } log⁡sin⁡(x/2)\log_{\sin(x/2)}{} sin x is equal to –

A

1

B

0

C

4

D

¼

Answer

1

Explanation

Solution

Let y = lim⁡x→0+\lim _ { x \rightarrow 0 ^ { + } } logsinxlogsin(x2)\frac{{logsin}x}{{logsin}\left( \frac{x}{2} \right)} = lim⁡x→0+\lim_{x \rightarrow 0^{+}} 2cot⁡xcot⁡x2\frac{\cot x}{\cot\frac{x}{2}}

= lim⁡x→0+\lim_{x \rightarrow 0^{+}} 2 tan⁡x/2x/2tan⁡xx\frac{\frac{\tan x/2}{x/2}}{\frac{\tan x}{x}} × x/2x\frac{x/2}{x} = 1