 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ia/paper-1/11d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

n^{2} + n - n = \frac{n}{n ^{2} + n + n} = \frac{1}{1 + 1/ n + 1} .

Since

1/ n \to 0

and the square root is continuous at

1

, the denominator tends to

2

. The reciprocal limit law established above is applicable because this limit is nonzero. Hence

n \to \infty lim (n^{2} + n - n) = \frac{1}{2} .

The limit is one half; rationalization exposes the finite difference hidden by cancellation.

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