This paper gives some connection between the theory of Toeplitz operator theory and function theory of the polydisk, and establishes a limit theorem for a finite sum of finite products of Toeplitz operators on the Hardy space in the polydisk. As a consequence, it shows that the product of six Toeplitz operators with pluriharmonic symbols is compact if and only if the product equals zero if and only if one of these Toeplitz operators equals zero.