= Divide-and-conquer asymptotic expansion
A <divide-and-conquer asymptotic expansion> splits an <integral> at an intermediate scale, expands each region where its approximation is uniform, and recombines the pieces. Terms depending on the arbitrary splitting point must cancel to the claimed error order. This is useful when an endpoint and the rest of the domain require different approximations.
For $0<\epsilon\ll1$, consider
$$
I(\epsilon)=\int_0^1\frac{dx}{x+\epsilon}=\log\frac{1+\epsilon}{\epsilon}.
$$
Choose $\epsilon\ll\delta\ll1$. On $[0,\delta]$, put $x=\epsilon X$ and integrate the inner form exactly:
$$
I_{\rm in}=\log(1+\delta/\epsilon)
=\log(\delta/\epsilon)+O(\epsilon/\delta).
$$
On $[\delta,1]$, replacing $(x+\epsilon)^{-1}$ by $x^{-1}$ gives
$$
I_{\rm out}=-\log\delta+O(\epsilon/\delta).
$$
Their sum is $-\log\epsilon+O(\epsilon/\delta)$, independent of $\delta$ at leading order. Taking $\delta=\sqrt\epsilon$ gives a vanishing error. Naively expanding the entire original <integrand> as $x^{-1}-\epsilon x^{-2}+\cdots$ fails at zero; the split makes the domain of each approximation explicit.
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