Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over $\mathcal{D}(\mathbb{R})$.
I've seem some times people talking about "taking a limit in the distributional sense". I wondered what this meant and came up with the following:
If we have a sequence $(f_n)$ with $f_n\in \mathcal{D}'(\mathbb{R})$, we may define pointwise convergence by saying that $(f_n)$ converges to $f\in \mathcal{D}'(\mathbb{R})$ if for each $\phi\in \mathcal{D}(\mathbb{R})$ we have that $(f_n(\phi))$ converges to $f(\phi)$ as a sequence of numbers.
This definition really says that $(f_n)$ converges to $f$ if
$$\lim_{n\to\infty}f_n(\phi)=f(\phi), \quad \forall \phi\in \mathcal{D}(\mathbb{R}). $$
Now in analogy to that I made the following definition:
Let $\psi : \mathbb{R}\to \mathcal{D}'(\mathbb{R})$, and let $a\in \mathbb{R}$. We say that the limit of $\psi(x)$ when $x$ goes to $a$ is $f\in \mathcal{D}'(\mathbb{R})$ if for every $\phi\in \mathcal{D}(\mathbb{R})$ we have
$$\lim_{x\to a}\psi(x)(\phi)=f(\phi),$$
that is I defined the limit in a pointwise fashion. In the particular case where $\psi(x)$ is a function in the usual sense, taking the limit this way is what I believe should be that "limit in the distributional sense".
Is that conclusion correct? Is this how we define limit of distributions and in particular how we define "limit in the distributional sense"?