Equivalent infinitesimal for $\log(\cos(x))$

Question

I recently came across a limit problem which replaces $\log(\cos(x))$ with an equivalent infinitesimal $\cos(x) - 1$.

How do we prove that $\cos(x)-1$ is the equivalent infinitesimal for $\log(\cos(x))$?

It would be good if you would "accept" the given answer, so this question does no more appear as "open" question. — Gottfried Helms, Jul 18 '18 at 08:00

score 4 · Accepted Answer · answered Jun 18 '18 at 17:56

4

Write

$$ \ln(\cos(x))=\ln(1+\cos(x)-1)=\cos(x)-1 +o(\cos(x)-1)$$

answered Jun 18 '18 at 17:56

Pagode

Of course!. Thank you and silly of me. – Sandeep Deb Jun 18 '18 at 18:05
You're welcome. Sillyness could appear in absent-mindedness of genius ;) – Pagode Jun 18 '18 at 18:12

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