Lorentzian contribution from shrinking regularization ($\int a[h(u{)}^2+a^2{]}^{-1}du\to \pi /|h^{\prime }(u_0)|$)

If $h$ has a simple interior zero $u_0$ and $a\to 0^{+}$, the inner variable $v=h^{\prime }(u_0)(u-u_0)/a$ turns the local integral into $(1/|h^{\prime }(u_0)|){\int }_{\mathbb{R}}(1+v^2{)}^{-1}dv$. Thus a peak with vanishing width can retain finite area. Slowly varying positive regularization factors cancel between peak height and width when the numerator has the matching factor. An ordinary outer expansion can miss this contribution even though the peak is exponentially narrow.