 Lorentzian peak approaching an integration endpoint

When $c(0)>0$ and $\lambda -1=O(ϵ)$, the change $x=ϵX$ resolves a truncated Cauchy distribution. Its leading contribution is $[ϵ\lambda c(0){]}^{-1}[\pi /2+\arctan ((\lambda -1)/(ϵc(0)))]$. It matches the nonresonant endpoint integral on one side and the full interior peak on the other. An interior peak at $x_{\ast }$ has width $ϵc(x_{\ast })/\lambda$; proximity to an endpoint must be measured against this width, not against $ϵ$ alone.