The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in C^{\infty}_{c}(\mathbb{R}\backslash \{0\}), $$
where $C^{\infty}_{c}((\mathbb{R}\backslash \{0\})= \{ f: (\mathbb{R}\backslash \{0\} \rightarrow \mathbb{C} \, ; \ f \ \ C^{\infty}, \, f \text{ has compact support in } \mathbb{R}\backslash \{0\} \ \} $.
To prove this I need to find a sequence of functions $\varphi_n \in C^{\infty}_{c}(\mathbb{R})$ with support contained in $\{ x; \, \frac{1}{n} \leq |x| \leq \frac{2}{n} \}$ satisfying $\varphi_n \rightarrow 0$ in $C^{\infty}_{c}(\mathbb{R})$ and $\langle u, \varphi_n \rangle \rightarrow + \infty$.
In fact, no, the method of proof is slightly different. You don't need to find a sequence of test functions that converges to zero in $C^\infty_c(\Bbb R)$.
Suppose the contrary: there exists a distribution $u\in D'(\Bbb R)$ such that it's restriction on $\Bbb R^\ast$ is represented by $\exp(1/x^2)$. Then, by definition of distribution, we can fix a compact set $K=[0,2]$ and find constants $C_K$ and $p_K$ such that $$|\langle u,\phi\rangle| \le C_K \sup_{j\le p_K}\|\phi^{(j)}\|_\infty$$for any test function $\phi$ with support contained in $K$.
Now that hints to proceed:
1) take $$g(x)=\begin{cases}0,&|x|\ge 1,\\e^{-\frac{1}{1-x^2}},&|x|\le 1.\end{cases}$$ Prove that this is a test function. What is its support?
2) take $h_n(x) = g\left(n\left(x-\frac{3}{2n}\right)\right)$. Find its support.
3) can you find a lower bound on $|\langle u,h_n\rangle| $? (hint: you're integrating a product of two continuous sign-constant functions) What's the type of dependence on $n$?
4) can you find an upper bound on $\sup_{j\le p_K}\|\phi^{(j)}\|_\infty$? What's the type of dependence on $n$?
5) conclusion.
If you have questions regarding any of the above, ask in comments.