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This paper is concerned with a phase field system of Cahn--Hilliard type that is related to a tumor growth model and consists of three equations in {\gianni terms} of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers \cite{CGH} and \cite{CGRS} {\gianni from} the viewpoint of well-posedness, long time \bhv\ and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in \cite{CGRS} by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates