 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/2/a/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

For fixed real

a

, the base is positive for all sufficiently large positive

x

ln (1 - u) = - u + O (u^{2})

ln [(1 - \frac{a}{x})^{x}] = x ln (1 - \frac{a}{x}) = - a + O (x^{- 1}) .

x \to \infty lim (1 - a / x)^{x} = e^{- a}

. This includes

a = 0

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