Let $u\in \mathcal{D}'(\mathbb{R}^d)$, $f\in C_c^\infty(\mathbb{R}^d)$ and $f(x)=1$ if $|x|\leq 1$; $f(x)=0$ if $|x|>2$. Can we get the following conclusion: there exists $v\in \mathcal{D'}(\mathbb{R}^d)$, such that, for any $\phi\in \mathcal{D}(\mathbb{R}^d)$,

$$ \lim_{n\to\infty} \langle u \ f(n\cdot),\phi\rangle = \langle v,\phi\rangle, $$ i.e. $u\ f(n\cdot) \stackrel{\mathcal{D'}(\mathbb{R}^d)}{\longrightarrow} v$.