# Using the Maclaurin series when calculating limits

There is a question I have about the Maclaurin series that I still find a bit tricky even though I know this is something I should have understood long ago.

Let's say we have a series convergent for any $$x$$, say $$e^x$$ or $$\sin{x}$$. It's quite a popular problem type to calculate limits where $$x \to 0$$ and using the Mclaurin expansion saves much time.

As far as I understand the Maclaurin series, it represents the function best in the neighborhood of $$x=0$$. This is why we should use the Maclaurin series for limits when $$x \to 0$$. But for anywhere convergent series, would it be wrong to use expansions at other points, like $$x=5$$, for example? Is it possible to get an incorrect limit value in that case?

You can use the expansion of Taylor-Maclaurin on finite points $$a$$ , however, it won't have the same expression because you need to apply it to $$f(x-a)$$ and not to $$f(x-0)$$. If you use the expansion in $$a$$ while you are in neighborhood of $$b$$, it makes no sense because higher order terms will be non negligible.