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Suppose I want to investigate the behaviour of say $\sin(\delta\ln(1+\epsilon))$ for variables $\delta$ and $\epsilon.$ I want to see what orders of $\delta$ and $\epsilon$ the term comes out as.

I thought I can expand $\ln$ by Taylor expansion. How many terms of that expansion should I take? If I take just say the first two terms, I'll get a different answer when I plug it into the Taylor expansion of $\sin$ than if I take three terms.

How do I know which is correct?

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For $\delta$ and $\epsilon$ much smaller than $1$ you are interested in the lowest order term in the Taylor series. If there is a term in $\delta^1$ it will be much larger than any term in $\delta^2$ as $\delta \to 0$. You just need to keep enough terms that you get a non-zero one.

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