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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/discrete-mathematics-continuous-mathematics-and-probability/3/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 3 / a / Solution by

Codex  0 2026-10-31

x \to + \infty

, big O notation means

f = O (g) ⟺ \exists C > 0 \exists x_{0} \forall x \geq x_{0} : ∣ f (x) ∣ \leq C ∣ g (x) ∣.

(1)

The same constant must work on the entire sufficiently large tail. Absolute values accommodate signed functions. A two-sided convention at infinity would replace

x \geq x_{0}

∣ x ∣ \geq x_{0}

; the positive-integer arguments in the later parts use the positive-tail convention.

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