Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 15A / e / i / Solution

For a test function $f$, substitute $y=nx$ and use the Taylor expansion $f(y/n)=f(0)+(y/n)f^{\prime }(0)+O(n^{-2})$. Oddness removes the first term, while the Gaussian integral gives $\int y^2{e}^{-y^2}dy=\sqrt{\pi }/2$. HenceTherefore