Solution
= Solution
For a test function $f$, substitute $y=nx$ and use the <Taylor expansion> $f(y/n)=f(0)+(y/n)f'(0)+O(n^{-2})$. Oddness removes the first term, while the <Gaussian integral> gives $\int y^2e^{-y^2}dy=\sqrt\pi/2$. Hence
$$
\int n^3xe^{-n^2x^2}f(x)\,dx\longrightarrow\frac{\sqrt\pi}{2}f'(0)
=\left\langle-\frac{\sqrt\pi}{2}\delta',f\right\rangle.
$$
Therefore
$$
\boxed{n^3xe^{-n^2x^2}\longrightarrow-\frac{\sqrt\pi}{2}\delta'(x).}
$$