Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that
$$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g \rangle| \leq \textrm{const}\|\tilde{g}\|_1, $$ where $\tilde{g}$ is Fourier transform of $g$ and $\|\tilde{g}\|_1:=\int_\mathbb{R} |\tilde{g}(\omega)| \, d\omega$.
What might be said about the existence of the limit $$ \lim_{a\rightarrow\infty} \langle T, g_a \rangle. $$ where $g_a(x):=g(ax)$.
The main difficulty, I think, comes form building such a distribution $T$. One can take $T=\delta_0$. In this case $\langle T,g_a\rangle=g(0)$ is a constant sequence.
Another example would be to take $T\in L^1(\Bbb R)$. In this case $\langle T,g_a\rangle\to0 $ as $a\to\infty$.
One of the examples of such distributions of first order is $p.v.(1/x)$, however, since this distribution is homogenous, $$\langle p.v.(1/x),g(ax)\rangle = \lim_{\varepsilon \to0}\int_{x\ge \varepsilon}\frac{g(ax)-g(-ax)}{x}dx=\lim_{\varepsilon \to0}\int_{y\ge a\varepsilon}\frac{g(y)-g(-y)}{y}dy = \langle p.v.(1/x),g(x)\rangle,$$ hence the sequence is once again constant.
It would be interesting to a) build a non-homogenous distribution $T$ of first order satisfying the estimations in terms of $\|\hat g\|_1$; b) build such a distribution $T$ of zeroth order such the limit of $\langle T,g_a\rangle$ does not exist (or show that it always converges).