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We discuss the problem of showing convergence of solutions when natural parameters that enter in the formulation of the Euler equations approach certain specific limits. More precisely, we start studying the free boundary Euler equations, showing that, if the boundary is sufficiently regular, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity. We also formulate a similar statement about the convergence of solutions of the compressible Euler equations to those of the incompressible ones, in which case the relevant parameter is the sound speed. Finally, if time allows, we shall discuss how the techniques employed in our analysis can also be used to prove well-posedness of the equations of fluid motion in all the above cases. This is a joint work with David G. Ebin.