= Distinguished limit
A <distinguished limit> rescales a parameter-dependent problem so that competing terms contribute together to its leading <dominant balance>. It identifies a transition regime rather than merely setting the small parameter to zero. For $\epsilon u''+u'=0$ with an order-one change across width $\delta$, the two terms scale as $\epsilon/\delta^2$ and $1/\delta$. Their ratio $\epsilon/\delta$ stays order one precisely when $\delta$ is proportional to $\epsilon$.
With $X=x/\epsilon$, the leading equation is $Y''+Y'=0$. A much broader scale discards diffusion; a much narrower one discards drift. Neither alone captures a bounded nonconstant transition from the endpoint value to the <outer solution>. The amplitude scale matters as well as the length scale, as the <one-third-power scaling at a nonlinear turning endpoint> illustrates.
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