We predict $s-$wave elastic cross-sections $sigma$ for low-energy atom-molecule collisions with kinetic energies up to 40 mK, for the $^4$He collision with weakly bound diatomic molecules formed by $^4$He with $^7$Li, $^6$Li and $^{23}$Na. Our scattering calculations are performed by using diatomic and triatomic molecular binding energies obtained from several available realistic models as input in a renormalized zero-range model, as well as a finite-range one-term separable potential in order to quantify the relevance of range corrections to our predictions. Of particular relevance for possible experimental realization, we show the occurrence of a zero in $sigma$ for the collision of cold $^4$He on $^4$He$-^{23}$Na molecule below 20 mK. Also our results for the elastic collision $^4$He on $^4$He$-^{6,7}$Li molecules suggest that $sigma$ varies considerably for the realistic models studied. As the chosen molecules are weakly bound and the scattering energies are very low, our results are interpreted on the light of the Efimov physics, which explains the model independent and robustness of our predictions, despite some sensitivity on the potential range.
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