lim-x-0-log-sin-x-cos-x-log-sin-x-2-cos-x-2-

Question Number 157123 by cortano last updated on 20/Oct/21

\lim_{x \to 0} \frac{\log_{\sin x} (\cos x)}{\log_{\sin (\frac{x}{2})} (\cos \frac{x}{2})} = ?

Commented by john_santu last updated on 21/Oct/21

L = \lim_{x \to 0} \frac{\ln (\cos x) \ln (\sin \frac{x}{2})}{\ln (\cos \frac{x}{2}) . \ln (\sin x)}

L = \lim_{x \to 0} (\frac{\ln (\cos x)}{\ln (\cos \frac{x}{2})}) . \lim_{x \to 0} (\frac{\ln (\sin \frac{x}{2})}{\ln (\sin x)})

L = \lim_{y \to 0} (\frac{\ln (\cos 2 y)}{\ln (\cos y)}) . \lim_{y \to 0} (\frac{\ln (\sin y)}{\ln (\sin 2 y)})

L = \lim_{y \to 0} (\frac{\frac{- 2 \sin 2 y}{\cos 2 y}}{\frac{- \sin y}{\cos y}}) . \lim_{y \to 0} (\frac{\frac{\cos y}{\sin y}}{\frac{2 \cos 2 y}{\sin 2 y}})

L = \lim_{y \to 0} (\frac{2 \tan 2 y}{\tan y}) . \lim_{y \to 0} (\frac{\cot y}{2 \cot 2 y})

L = 4 \times \lim_{y \to 0} (\frac{\tan 2 y}{2 \tan y}) = 4 \times 1 = 4