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Precise Meaning of f(n) = 1/n → 0

publish date: 2026/05/23 22:07:24.303698 UTC

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The statement "\(f(n) = 1/n\) tends to 0 as \(n \to \infty\)" means precisely:

1/n = 0 for n = ∞

For any positive ε, however small, we can find N such that 1/n < ε whenever n > N

1/n = 0 for all n > 1000

1/n decreases and eventually becomes negative

Correct Answer

For any positive ε, however small, we can find N such that 1/n < ε whenever n > N

Explanation

The precise meaning: for any \(\varepsilon > 0\) (no matter how tiny), there exists \(N\) such that \(n > N \Rightarrow 1/n < \varepsilon\). Simply take \(N = \lceil 1/\varepsilon \rceil\). This is the \(\varepsilon\)–\(N\) definition of the limit \(\lim_{n \to \infty} 1/n = 0\) — the foundation of all calculus.

Reference

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012

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