 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/1/b/solution

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

sin x - arctan x = x^{3} /6 + O (x^{5})

, while

x^{2} ln (1 + x) = x^{3} (1 - x /2 + O (x^{2}))

. Dividing the two series gives

\frac{sin x - arctan x}{x ^{2} ln ( 1 + x )} = \frac{1}{6} + O (x) .

The first nonzero term after cancellation is the constant

1/6

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