When is the phase shift small and linear in $k$ for low energy $s$-wave scattering?

I was wondering when should we expect the phase shift to be small and linear in $k$ for low energy $s$-wave scattering, so that the scattering length can defined as $-delta/k$, since the general form of the scattering length for low energy is $-tan(delta)/k$, the two only seem to be equal to each other when the phase shift is small or close to zero. Also will a scattering length that is equal to $-delta/k$ always be finite and not approach infinity like $-tan(delta)/k$ would when $delta=pi/2$.