 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-3/15a/e/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

f

, substitute

y = n x

f (y / n) = f (0) + (y / n) f^{'} (0) + O (n^{- 2})

. Oddness removes the first term, while the Gaussian integral gives

\int y^{2} e^{- y^{2}} d y = π /2

. Hence

\int n^{3} x e^{- n^{2} x^{2}} f (x) d x ⟶ \frac{π}{2} f^{'} (0) = ⟨ - \frac{π}{2} δ^{'}, f ⟩ .

Therefore

n^{3} x e^{- n^{2} x^{2}} ⟶ - \frac{π}{2} δ^{'} (x) .

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