Estimating rate-induced tipping via asymptotic series and a Melnikov-like method*\n *CK was partly supported by a Lichtenberg Professorship of the VolkswagenStiftung. CK also acknowledges partial support of the EU within the TiPES project funded the European Unions Horizon 2020 research and innovation programme under Grant Agreement No. 820970. IPL was partly supported by MICIIN/FEDER project RTI2018-096523-B-100 and by European Union's Horizon 2020 research and innovation programme under the Marie Sk0̆142odowska-Curie Grant Agreement No. 754462. CK and IPL also want to thank Hildeberto Jardon-Kojakhmetov for interesting discussions regarding asymptotic analysis in the early phase of this project.\n