  We examine the correspondence between QFT observables and bulk solutions in
the context of AdS/CFT in the limit as the cosmological constant $\Lambda \to
0$. We focus specifically on the spacetime metric and a non-backreacting scalar
in the bulk, compute the one-point functions of the dual operators and
determine the necessary conditions for the correspondence to admit a
well-behaved zero $\Lambda$ limit. We discuss holographic renormalization in
this limit and find that it requires schemes that partially break
diffeomorphism invariance of the bulk theory. In the specific case of three
bulk dimensions, we compute the zero $\Lambda$ limit of the holographic Weyl
anomaly and reproduce the central charge that arises in the central extension
of $\mathfrak{bms}_{3}$. We compute holographically the energy and momentum of
those QFT states dual to flat cosmological solutions and to the Kerr solution
and find an agreement with the bulk theory. We also compute holographically the
renormalized 2-point function of a scalar operator in the zero $\Lambda$ limit
and find it to be consistent with that of a conformal operator in two
dimensions less. Finally, our results can be used in a new definition of
asymptotic Ricci-flatness at null infinity based on the zero $\Lambda$ limit of
asymptotically Einstein manifolds.
