= Matched cutoff estimate for a narrow Lorentzian integral
{title2=$\int_0^\infty e^{-x}/(\epsilon^2+x^2)\,dx\sim\pi/(2\epsilon)$}
Split the <integral> at $\epsilon\ll\delta\ll1$. Replacing the numerator by one below the cutoff gives $\epsilon^{-1}\arctan(\delta/\epsilon)$, with an error bounded by $\tfrac12\log(1+\delta^2/\epsilon^2)$. Above the cutoff, replacing the denominator by $x^2$ has relative error $O(\epsilon^2/\delta^2)$, and <integration by parts> gives $1/\delta+\log\delta+O(1)$. The inner $-1/\delta$ cancels the outer $1/\delta$, and all errors are $o(1/\epsilon)$, proving the leading constant independently of the intermediate cutoff.
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