Given $F_n\in C^\infty_0$ given by $F_n(\varphi)=\left(\varphi\left(\frac{1}{n}\right)-\varphi\left(-\frac{1}{n}\right) \right)$.
Then what is the distribution given by $\lim_n F_n?$.
The problem lies in how the limit will behave, because it doesn't looks there is a limit at all.
As @RideTheWavelet says, for every test function $\varphi$, the limit of $F_n(\varphi)$ is $0$, because of continuity at $0$ of such $\varphi$. Thus, by the definition of the weak dual topology on distributions, $\lim_n F_n=0$ (where limit is taken in that topology).