The limitations of the Taylor series approximations of the delayed variables have been documented by means of examples in several works. This note unifies these previous comments through a higher level analysis of the issue. First, a general negative result is derived for the use of higher order truncation of the Taylor series. It covers the properties of the special examples in the literature. Second, it is shown that the use of the truncated Taylor series expansion of the (inverse) advance operator does not feature similar drawbacks and may be efficient in practice, depending on the system parameters.