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

$$\underset{x\to 0}{\lim }\frac{\log {}_{\sin x}\left(\cos x\right)}{\log {}_{\sin \left(\frac{x}{2}\right)}\left(\cos \frac{x}{2}\right)}=?$$

Commented by john_santu last updated on 21/Oct/21

$$L=\underset{x\to 0}{\lim }\frac{\ln \left(\cos x\right)\ln \left(\sin \frac{x}{2}\right)}{\ln \left(\cos \frac{x}{2}\right).\ln \left(\sin x\right)}$$$$L=\underset{x\to 0}{\lim }\left(\frac{\ln \left(\cos x\right)}{\ln \left(\cos \frac{x}{2}\right)}\right).\underset{x\to 0}{\lim }\left(\frac{\ln \left(\sin \frac{x}{2}\right)}{\ln \left(\sin x\right)}\right)$$$$L=\underset{y\to 0}{\lim }\left(\frac{\ln \left(\cos 2y\right)}{\ln \left(\cos y\right)}\right).\underset{y\to 0}{\lim }\left(\frac{\ln \left(\sin y\right)}{\ln \left(\sin 2y\right)}\right)$$$$L=\underset{y\to 0}{\lim }\left(\frac{\frac{-2\sin 2y}{\cos 2y}}{\frac{-\sin y}{\cos y}}\right).\underset{y\to 0}{\lim }\left(\frac{\frac{\cos y}{\sin y}}{\frac{2\cos 2y}{\sin 2y}}\right)$$$$L=\underset{y\to 0}{\lim }\left(\frac{2\tan 2y}{\tan y}\right).\underset{y\to 0}{\lim }\left(\frac{\cot y}{2\cot 2y}\right)$$$$L=4\times \underset{y\to 0}{\lim }\left(\frac{\tan 2y}{2\tan y}\right)=4\times 1=4$$

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