Let $f_n : \mathbb{R} \to \mathbb{C}$ $$\begin{align}&f_n(x) = n \sin n \pi x \quad 0 \leq x \leq 1 \\ &f_n(x) = 0 \quad x \lt 0 \lor x \gt 1\end{align}$$
- Can you calculate Fourier transforms of $f_n$?
- Can you calculate first derivatives of $f_n$?
- Can you tell what distribution do $f_n$, $f_n'$ and Fourier transforms tend to as $n \to \infty$?
Can you? :)
The transform of $f_n$ can be computed directly by converting the sine to complex exponentials.
In the image $F_n$, replace $n^2/(n^2-w^2/\pi^2)$ by $1$ to get $\widetilde F_n$.
Then it can be proved that the distributional limit of $F_n - \widetilde F_n$ is zero. $\widetilde F_n$ will contain a constant term and a $(-1)^n e^{i w}$ term, for which reason we cannot just take the limit of $F_n$ directly.
The inverse transform of $\widetilde F_n$ will give a $\delta(x)$ term and a $(-1)^n \delta(x-1)$ term, omitting constant factors.