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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?

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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.

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