Part 1
I don't understand a part of my lectures on convergence of distributions. We are proving that principal value distribution ($\operatorname{vp}\frac{1}{x}$) is not a distribution of order $0$. We have defined a sequence of functions: $\phi_{n}(x)=\rho (x_0+nx)$, where $\rho$ is the standard Friedrichs' mollifier. We suppose the opposite; that it is of order $0$. Then, $\langle \operatorname{vp}(\frac{1}{x}), \phi_{n} \rangle \le C\lvert \lvert \phi_{n} \rvert \rvert _{L^{\infty}}=C\lvert \lvert \rho \rvert \rvert _{L^{\infty}}=D=\mathrm{const}.$ Then we conclude this:
$$\left\langle \operatorname{vp}\left(\frac{1}{x}\right), \phi_{n} \right\rangle =\langle (\ln(\lvert x \rvert))',\phi_{n} \rangle = \langle (\ln(\lvert x \rvert),\phi_{n}'\rangle= \lvert \int \ln(\lvert x \rvert) \rho_{n}^{x_0}(x)dx \rightarrow \lvert \ln(\lvert x_0 \rvert) \rvert .$$
I don't understand the last conclusion. I know that $\rho_{n}\rightarrow \delta_0$ and $\rho_{n}^{(n)}\rightarrow \delta_0^{(n)}$ but this looks like we concluded that $\phi_{n}'\rightarrow \delta_{x_0}$.
I need to understand this argument as I use it to show that Partie finie ($Pf \frac{1}{x^2} $) is of order $2$ with the exact sequence of functions as counterexample in case it's of order $1$.
Part 2:
For, $f_{n}(x)=n^2 x\cos(nx)$, is $\lim _{n\rightarrow \infty}f_{n}(x)=0$? I've shown it using partial integration three times, but I'm not sure whether each step is justified.