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We investigate the dependence of the center-of-mass tomogram of a system withmany degrees of freedom $N$ on the Planck constant $\hbar $. It is shown thatto use the central limit theorem under taking the limit $N\to +\infty $ oneshould keep the energy of the system to be constant. In the case, the resultingdistribution is Gaussian if the initial distribution is a product ofindependent excited states of a quantum oscillator or even and odd coherentstates either. Then, if one turns the Planck constant $\hbar \to 0$ we get$\delta $-function associated with the distribution concentrated in zero withthe probability equal to one.

A use of the central limit theorem to obtain a classical limit for the center-of-mass tomogram