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We consider a sequence of functions which are starlike in the unit disk and their logarithmic derivatives are meromorphic with a finite number of simple poles in any boundary domain. These poles are either determined or random with given characteristics. The aim of the article is the limit process and the properties of limit functions. We distinguish conditions for residues and distribution of poles. Under certain conditions, the sequence converges to the identity function. Another conditions allow us to obtain estimates for the limit function and its logarithmic derivative.