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!
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.