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MathidiaMathematics Academy
Sequences and series
Lesson 26 of 75
Introduction

Limits of sequences

Objective. Prove and calculate sequence limits.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A sequence limit describes the single value that the terms of a sequence approach as the index grows without bound. Formally, aₙ → L means for every ε > 0 there exists N such that |aₙ − L| < ε whenever n > N, capturing "eventually arbitrarily close". Limit laws let you combine known limits: if aₙ → L and bₙ → M then aₙ + bₙ → L + M and aₙbₙ → LM. The squeeze theorem is a key tool: if aₙ ≤ bₙ ≤ cₙ and both outer sequences tend to L, then bₙ → L too. For example, since −1/n ≤ (−1)ⁿ/n ≤ 1/n and both bounds tend to 0, (−1)ⁿ/n → 0.

Worked example

Find the limit of the sequence aₙ = (3n + 2)/(n + 5) as n → ∞.

3 … 2
Illustrated card showing 3 … 2
  1. 1Divide numerator and denominator by n: aₙ = (3 + 2/n)/(1 + 5/n).
  2. 2As n → ∞, 2/n → 0 and 5/n → 0 by the reciprocal limit law.
  3. 3Substitute these limits: aₙ → (3 + 0)/(1 + 0).
  4. 4Simplify the fraction to get the limit.

Answer: 3

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Common mistake. Students assume a bounded sequence must converge, citing 1, −1, 1, −1, … as "close to a limit". Boundedness alone is not enough; always check the terms actually settle near one value using the ε–N definition.

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