The limit of an infinite sequence (a_k) of real numbers a_k is determined solely by the finite terms of the sequence. Otherwise, the limit would not have to be *computed* but would have to be *created*. Analysis is concerned with analyzing, i.e., with finding.
To give an example, we can state with absolute certainty that in the real numbers the sequence 0.1, 0.11, 0.111, ... has the limit 0.111... = 1/9. That is independent of the method which is used to analyze the sequence. But there are different aspects of the limit, namely the numerical value of the limit, the set of coefficients of the power series, its cardinal number, the set of indexes which belong to a digit 1, its cardinal number, the set of indexes which belong to a digit 2, its cardinal number, the set of different digits appearing in the limit, and many further aspects.
If any of these aspects is computed by another than the analytical method and turns out as deviating from the analytical result, then the other method is not suitable for analytical purposes.