The authors show that the solution of the initial value problem for the compressible Navier-Stokes-Korteweg system \(\partial_t\rho+\nabla\cdot(\rho u)=0\), \(\partial_t u+(u\cdot\nabla)u+\nabla P(\rho)/\rho=(\mu/\rho)\triangle u+ ((\mu+\nu)/\rho)\nabla(\nabla\cdot u)+\kappa \nabla\triangle\rho\), \((\rho,u)(x,0)=(\rho_0,u_0)(x)\) converges to the solution of the system if the capillary constant \(\kappa\) is zero, which is the compressible Navier-Stokes system. Notations: \(x\in{\mathbb R}^n\), \(t\geq 0\), \(\rho>0\) density, \(u\) velocity, \(P(\rho)\) pressure function, \(\mu\) and \(\nu\) viscosity coefficients satisfying \(\mu>0\), \(2\mu+3\nu\geq0\), and \(\kappa>0\) is the capillary constant. The convergence is on every fixed time interval \([0,T]\), \(T<\infty\), and with respect to appropriate Banach space norms. The proof is based on a series of a priori estimates, on a Sobolev space interpolation inequality of Gagliardo and Nirenberg and on a compactness argument due to Lions and Aubin.