Let $\phi\in \mathcal D(\mathbb R)$ be a test function. Let $$ u_m(x)=m\sin (mx^2) $$ be a sequence of functions that defines a sequence of distributions in $\mathcal D'(\mathbb R)$. What is the limit of $u_m$ in $\mathcal D'(\mathbb R\backslash\{0\})$?
I think it might be a good idea to compare this to the sequence $v_m(x)=m\sin (mx)$. After all, for large $x$ and "small" $\text{supp} \phi$, we can assume $x^2$ has constant gradient inside $\text{supp} \phi$, so the situation is very similar to $m\sin (mx)$. However, the limit of $v_m$ doesn't seem to exist. In fact, the value of the integral $$ \int_1^2 m\sin(mx^2)dx $$ oscillates between $-2$ and $2$, or something like that. However, $\chi_{[1,2]}$ is not a test function, so I am not quite sure.
The best thing I can prove for $u_m(x)$ is that the value of $\langle u_m,\phi\rangle$ is a bounded sequence for all $\phi\in \mathcal D(\mathbb R)$.
So, what are the limits of $u_m$ and $v_m$ in $\mathcal D'(\mathbb R)$?
$g_m(y)=\sin(my)/m$, it $\to 0$ uniformly thus in the sense of distributions
whence $g_m''\to 0$ in the sense of distributions
Since $x\to x^2$ is $C^\infty$ with $C^\infty$ inverse on $(0,\infty)$ and $(-\infty,0)$ then $f_m(x)=g_m''(x^2) \to 0$ in the sense of distributions on $\Bbb{R}^*$