Limit of a Sequence: Definition, Computation and Properties

A sequence (a_n) converges to the limit L if the terms get arbitrarily close to L as n grows: for every ε > 0 there is an N such that |a_n − L| < ε for all n > N. In brief: the sequence a_n = (3n + 1)/(n + 2) tends to 3. To compute such limits, divide the numerator and the denominator by the highest power of n: (3 + 1/n)/(1 + 2/n) → 3/1. A sequence that does not converge diverges.

The limit of a sequence is the foundation of calculus: derivatives, integrals and series are all defined by limits. The formal definition may look abstract, but it captures a simple idea. This guide explains the definition with a worked proof, the limit laws, how to compute limits of rational and root expressions, the standard limits, the concepts of divergence and boundedness and the relation to monotone sequences, with a table of limits.

What is the definition of a limit?

We write lim_n→∞ a_n = L (or a_n → L) if for every ε > 0 there exists a natural number N such that |a_n − L| < ε for all n > N. In words: for any tolerance ε, all terms from some index on lie in the ε-neighborhood of L. Only finitely many terms may lie outside. A limit, if it exists, is unique.

Example 1: a proof from the definition

Show that 1/n → 0. Given ε > 0, choose N ≥ 1/ε. For n > N, |1/n − 0| = 1/n < 1/N ≤ ε ✓. For ε = 0.01, N = 100: all terms after the 100th are below 0.01.

Example 2

Show that (2n + 1)/(n + 3) → 2. The difference is |(2n + 1)/(n + 3) − 2| = |(2n + 1 − 2n − 6)/(n + 3)| = 5/(n + 3). This is less than ε when n > 5/ε − 3. Choose N ≥ 5/ε. For ε = 0.1: N = 50. Check n = 51: (103)/(54) = 1.907, and 2 − 1.907 = 0.093 < 0.1 ✓.

What are the limit laws?

If a_n → A and b_n → B, then:

• a_n ± b_n → A ± B,
• c · a_n → cA for a constant c,
• a_nb_n → AB,
• a_n/b_n → A/B, if B ≠ 0 (and b_n ≠ 0),
• a_n^k → A^k and ^k√a_n → ^k√A (for A ≥ 0 when k is even),
• if a_n ≤ b_n for all n, then A ≤ B (strict inequalities become non-strict in the limit).

How do you compute limits?

Rational expressions

Divide the numerator and the denominator by the highest power of n in the denominator.

Example 3

(5n² − 3n + 1)/(2n² + 7) = (5 − 3/n + 1/n²)/(2 + 7/n²) → 5/2. And (n + 4)/(n² + 1) → 0, since the denominator has the higher degree. And (n³ + 1)/(n² + 5) → ∞, since the numerator has the higher degree.

Expressions with roots: multiply by the conjugate

Example 4

√(n² + n) − n = [(n² + n) − n²]/(√(n² + n) + n) = n/(√(n² + n) + n) = 1/(√(1 + 1/n) + 1) → 1/2.

Geometric sequences

rⁿ → 0 for |r| < 1; → ∞ for r > 1; for r = 1 it equals 1; for r ≤ −1 it diverges. Example: (2ⁿ + 3ⁿ)/3ⁿ = (2/3)ⁿ + 1 → 1. And (4ⁿ − 3 · 2ⁿ)/(4ⁿ + 1) → 1.

Sums

1 + 1/2 + 1/4 + … + 1/2ⁿ = 2 − 1/2ⁿ → 2. And (1 + 2 + … + n)/n² = n(n + 1)/(2n²) = (1 + 1/n)/2 → 1/2.

The number e

(1 + 1/n)ⁿ → e ≈ 2.71828. More generally (1 + x/n)ⁿ → e^x, and (1 − 1/n)ⁿ → 1/e.

What does divergence look like?

• Tending to infinity: a_n → ∞ if for every M there is N with a_n > M for n > N (n², 2ⁿ).
• Oscillation: (−1)ⁿ has two accumulation points, −1 and 1, so no single limit.
• A convergent sequence is bounded. So an unbounded sequence cannot converge.
• If a_n → L, then every subsequence tends to L. Two subsequences with different limits prove divergence.

Example 5

a_n = sin(nπ/2): the values 1, 0, −1, 0, 1, … The subsequence with n = 4k + 1 has the limit 1, and with n = 4k + 3 has the limit −1. So the sequence diverges.

What is the monotone convergence theorem?

Every monotone and bounded sequence converges. An increasing sequence bounded above converges to its supremum, and a decreasing sequence bounded below converges to its infimum. This is used to prove the existence of a limit without knowing its value.

Example 6

a₁ = 1, a_n+1 = √(2 + a_n): 1, 1.732, 1.932, 1.983, … The sequence is increasing and bounded by 2 (if a_n < 2 then a_n+1 < √4 = 2), so it converges. The limit L satisfies L = √(2 + L), so L² = 2 + L, i.e., (L − 2)(L + 1) = 0, and L = 2.

Table of limits

Where does the definition come from?

Ideas of limit appear in the method of exhaustion of Eudoxus and Archimedes. In the seventeenth and eighteenth centuries, Newton and Leibniz used infinitely small quantities without precise definitions. The rigorous ε–N definition was developed by Bernard Bolzano and Augustin-Louis Cauchy in the early nineteenth century, and made fully precise by Karl Weierstrass, who introduced the ε–δ language that is used today.

Where are limits of sequences used?

Numerical methods generate sequences that converge to the solution: Newton’s iteration x_n+1 = (x_n + 2/x_n)/2 tends to √2. Compound interest with n compounding periods tends to the continuous growth formula. In probability, the law of large numbers states that the average of independent trials tends to the expected value.

Which mistakes are common?

Substituting infinity as a number

∞ is not a number. Divide by the highest power instead of writing ∞/∞.

Assuming that terms close to a value mean convergence

The terms must stay within ε for all large n, not just for some. The terms of (−1)ⁿ are always close to ±1, but the sequence does not converge.

Applying the limit law for quotients when the denominator tends to 0

The rule a_n/b_n → A/B requires B ≠ 0. The case 0/0 needs algebra.

Forgetting that ∞ − ∞ is indeterminate

√(n² + n) − n looks like ∞ − ∞ but tends to 1/2. Transform the expression first.