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Let be $f:\mathbb{R}\to\mathbb{R}$ a $k$-times differentiable function and $T_k(x)$ its Taylor polynomial of $k$-th order about point $a$.

If we are closer to $a$ then the approximation gets better: $$ \lim\limits_{x\to a}\frac{f(x)-T_k(x)}{(x-a)^k}=0. $$

To show this limit one could simply apply $k$-times rule of L'Hopital.

However, I would like to know if there is another way (without rule of L'Hopital) to prove this limit?

Philipp
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  • If you assume that the $k$th derivative is continuous, then you can use the integral form of the remainder to prove this. Basically, any of the various remainder formulas can help you prove your result easily. – Mason Feb 22 '22 at 22:43
  • This follows easily by the mean-value theorem and induction on $k$ (recall that $k=1$ is true by definition of differentiability of $f$ at $a$; at the induction step, use the mean-value theorem). See here for the details listed out in the higher dimensional case (I leave it to you to see how everything is simplified in the one-dimensional case). – peek-a-boo Feb 22 '22 at 22:58

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