In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}, \ k\in Z\},\ n\ge 1,$ is a series of linear processes with innovations having heavy-tailed tapered distributions with tapering parameter $b_n$ depending on $n$. It is shown that, depending on the properties of a filter of a linear process under consideration and on the parameter $b_n$ defining if the tapering is hard or soft, the limit process for such partial sum process can be fractional Brownian motion or linear fractional stable motion.