= Solution
For an interaction of finite range $b$, higher <partial waves> are suppressed at $kb\ll1$ by the <centrifugal barrier>, away from exceptional higher-wave resonances. Outside the potential, the zero-energy radial <S-wave scattering> equation is $u''=0$, where $\psi=u/r$. Its solution is proportional to $r-a$, so normalization of the constant term gives
$$
\boxed{\psi\simeq1-\frac ar}.
$$
The <scattering length> $a$ is the extrapolated zero of $u(r)$. At nonzero small $k$, this form applies in the overlap region $b\ll r\ll k^{-1}$; it is not the far-field limit at fixed $k$. Equivalently, $k\cot\delta_0\to-1/a$ and $f_0\to-a$.
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