 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/5d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

δ_{ε} (x) = \frac{ε}{π ( ε ^{2} + x ^{2} )} .

δ

, and

g_{ε} = δ_{ε}^{'}

. For a smooth rapidly decaying test function

ϕ

\int_{- \infty}^{\infty} ϕ (x) g_{ε} (x) d x = - \int_{- \infty}^{\infty} ϕ^{'} (x) δ_{ε} (x) d x ⟶ - ϕ^{'} (0) .

⟨ δ^{'}, ϕ ⟩ = - ϕ^{'} (0)

. Therefore the limit as a distribution is

g_{ε} ⟶ δ^{'} .

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