Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$. Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$.
I'm reading the solution, and I understand how to prove that all derivates must be of the form: 
$$f^{(n)}(x)=e^{-1/x}\left[\sum_{k=0}^{2n} \frac{a_k}{x^k}\right]$$
After this, the solution manual begins to prove $f^{(n)}(0)=0$ for $n\in \Bbb N$ by induction. 

Assume that $f^{(n)}(0)=0$ for some $n\geq 0$. We need to prove
  $$\lim_{x\to 0} \frac{f^{(n)}(x)-f^{(n)}(0)}{x-0}=0$$

I don't understand why they do this. Isn't easier to just prove that for $n\in \Bbb N$, $f^{(n)}(x) \to 0$. I would think that using l'hopitale this must be possible.
 A: The formula you give for $f^{(n)}(x)$ is valid for $x > 0$.  So the problem is that it is not necessarily true that $f^{(n)}(0) = \lim_{x\to 0^+} f^{(n)}(x)$, as this would be assuming that the derivative was continuous.
A: In the theory of differentiable manifolds, the function
\begin{equation}\label{part-unity-function}\tag{#}
f(t)=
\begin{cases}
\textrm{e}^{-1/t}, & t>0\\
0, & t\le0
\end{cases}
\end{equation}
plays an indispensable role in the proof of the existence of partitions of unity. In the paper [1] below, among other things, some properties of the function $f(t)$ defined by \eqref{part-unity-function} were investigated.

*

*For $i\in\mathbb{N}=\{1,2,\dotsc\}$,
\begin{equation}\label{f(t)-derivative}
f^{(i)}(t)=
\begin{cases}\displaystyle
\frac{\textrm{e}^{-1/t}}{t^{2i}} \sum_{k=0}^{i-1}(-1)^k{k!}\binom{i}{k}\binom{i-1}{k}{t^{k}}, & t>0;\\
0, & t\le0.
\end{cases}
\end{equation}
See Theorem 2.3 in the paper [1] below.

*The function $f(t)$ defined by \eqref{part-unity-function} is infinitely differentiable on $\mathbb{R}$ and
\begin{equation}\label{f-der-at0=0}
f^{(i)}(0)=0
\end{equation}
for all $i\in\{0\}\cup\mathbb{N}$, but it is not analytic at $t=0$. See Theorem 2.4 in the paper [1] below.

By the way, at the site https://math.stackexchange.com/a/4262498/945479, there are discussions on the functions $\textrm{e}^{\pm1/t}$.
References

*

*Xiao-Jing Zhang, Feng Qi, and Wen-Hui Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, International Journal of Open Problems in Computer Science and Mathematics 5 (2012), no. 3, 122--127; available online at https://doi.org/10.12816/0006128.

