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Правило Лопиталя:

$\lim\limits_{x \to 0}\frac{ln(cos3x)}{ln(cosx)}=\lim\limits_{x \to 0}\frac{\frac{-3\, sin3x}{cos3x}}{\frac{-sinx}{cosx}}=\lim\limits_{x \to 0}\frac{3\, sin3x\cdot cosx}{cos3x\cdot sinx}=\\\\=\Big [\, sin3x\sim 3x\; ;\; sinx\sim x\; \; pri\; \; x\to 0\, \Big ]=\\\\=\lim\limits _{x \to 0}\frac{3\cdot 3x\cdot cosx}{x\cdot cos3x}=\lim\limits _{x \to 0}\frac{3\cdot 3x\cdot 1}{x\cdot 1}=\frac{9}{1}=9$