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?