Taylor's series around zero of $e^{-1/x^2}$?

Question

Consider the following piecewise function:

$f(x)=e^{-1/x^2}$ if $x \neq 0$, and $f(x)=0$ if $x=0$.

I'm trying to write her Taylor's series around $x=0$ but I'm stuck because calculating the derivative of $f(x)$ and evaluating at $x=0$ is not so straightforward, due to the evaluation needs to take the limit $x->0$.

Questions: 1. Which formal criteria or theorem can I use to justify that $f^{(n)}(0)=\lim_{x->0}f^{(n)}(x)$? 2. Is there a direct way of writing the complete Taylor's series without computing a lot of derivatives and limits?

Answer 1

By the MVT you show that if for $k\geq 1$:

• $f\in C^{k-1}$ in a neighborhood $U$ of 0,
• $f\in C^k$ in the punctured neighborhood $U\setminus\{0\}$ and
• $\ell=\lim_{x\rightarrow 0} f^{(k)}(x)$ exists then $f\in C^k(U)$ and $f^{(n)}(0)=\ell$.

In the present context you would usually first show that $\lim_{x\rightarrow 0} \frac{P(x)}{x^m} e^{-1/x^2}=0$ for any polynomial $P$, and any $m\geq 1$ and then use induction and the above statement.