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I am trying to learn about using Taylor to find limits as an alternative for my final exam. But one thing that confused me about the concept is when exactly do I stop expanding the functions? Since we know one can approximate a function with infinity terms with the error of approximation decreasing each time we increase an expression. Is there a thumb rule as to how many terms I consider or an easy to grasp non-formal mathematical intuition would be appreciated.

And can I use Taylor to lessen my work of test of convergence on improper integrals? Thanks!

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  • $\begingroup$ It depends on what type of limit you are trying to find, and how you are using Taylor expansions to find it. Also remember that Taylor series have radii of convergence, so in general they are not useful for limits at infinity. $\endgroup$ Jan 26, 2022 at 1:42

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To question 1. I consider a limit at point $-\infty<x_0<\infty$ of function $f(x)$, i.e. $\lim_{x\to x_0} f(x)$. Usually you cannot directly insert $x_0$ to calculate $f(x_0)$, because you would get an indeterminate term (like $\frac{0}{0}$). If you insert the Taylor series, usually only the lowest order term is needed, which if canceled, direct evaluation $f(x_0)$ is available. You need to do cancellation so many times (usually once is enough) so that you get a determinate term. To question 2: yes, and also here only the lowest order term is decisive. If you evaluate more terms then needed, you also get the right answer, but with more effort than needed.

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