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Question Number 109067 by n0y0n last updated on 20/Aug/20

Answered by Aziztisffola last updated on 20/Aug/20

$$\sin \left(\mathrm{x}\right)\ln \left({\cot }^2\left(\mathrm{x}\right)\right)=\sin \left(\mathrm{x}\right)\ln \left(\frac{{\cos }^2\left(\mathrm{x}\right)}{{\sin }^2\left(\mathrm{x}\right)}\right)$$$$=\sin \left(\mathrm{x}\right)(\ln \left({\cos }^2\left(\mathrm{x}\right)\right)-\sin \left(\mathrm{x}\right)\ln \left({\sin }^2\left(\mathrm{x}\right)\right)$$$$\{\begin{array}{l}\underset{\mathrm{x}\to 0}{\lim }\sin \left(\mathrm{x}\right)(\ln \left({\cos }^2\left(\mathrm{x}\right)\right)=0\\ \underset{\mathrm{x}\to 0}{\lim }\sin \left(\mathrm{x}\right)\ln \left({\sin }^2\left(\mathrm{x}\right)\right)=\underset{\mathrm{x}\to 0}{\lim 2sin}\left(\mathrm{x}\right)\ln \left(\sin \left(\mathrm{x}\right)\right)=0\end{array}$$$$\underset{\mathrm{x}\to 0}{\lim }\sin \left(\mathrm{x}\right)\ln \left({\cot }^2\left(\mathrm{x}\right)\right)=0$$

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