  Considering finite particle systems, we elaborate on various ways to pass to
the limit as thenumber of agents tends to infinity, either by mean field limit,
deriving the Vlasov equation,or by hydrodynamic or graph limit, obtaining the
Euler equation. We provide convergenceestimates. We also show how to pass from
Liouville to Vlasov or to Euler by taking adequatemoments. Our results
encompass and generalize a number of known results of the literature.As a
surprising consequence of our analysis, we show that sufficiently regular
solutions of anylinear PDE can be approximated by solutions of systems of N
particles, to within 1/ log log(N ).
