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0 $begingroup$ I would like to show that : $$lnleft(cos frac{1}{2^n}right) = Oleft(frac{1}{4^n}right)$$ Attempt : let's show that the limit : $limlimits_{n to infty} 4^nlnleft(cos frac{1}{2^n}right)$ is bounded. Now the problem is that I need to control at which speed $ln(...)$ go to $0$ . Yet I don't se how to do so. Thank you ! real-analysis calculus asymptotics share | cite | improve this question edited Jan 7 at 22:52 rtybase 10.7k 2 15 33 asked Jan 7 at 21:46