We study the relative Gromov-Witten theory on TP^1 \times P^1 and show that certain equivariant limits give us the relative invariants on P^1\times \P^1. By formulating the quantum multiplications on Hilb(TP^1) computed by Devash Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion and tangency operators computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits. Speaker(s): Shuai Wang (Columbia University)