The sequence does seem to have a limit as a continuous function. In the first example, it approaches the function, y = sin(x). This is because the first table forms a list of functions that are the Taylor series functions of sin(x).
The sequence “converges” in the sense that as there are more terms of the Taylor series for sin(x), it appears more and more like sin(x).
We could analyze the convergence by observing |sin(x) – fn(x)| as n goes to infinity. This is a list of functions of the Taylor series for sin(x), so as n approaches infinity, |sin(x) – fn(x)| will approach 0.