Limit of a Sequence

A sequence is an endless line of numbered values, one for each n. The real question is whether those values settle toward a single destination as n grows larger and larger, and that destination is what we call the limit. If the terms crowd arbitrarily close to one number we say the sequence converges; if they scatter or bounce forever, it diverges. Drag the term-count slider to watch the dots close in on L = 1, then switch between sequences to compare convergence against divergence side by side.

Where the Sequence Heads

Terms to show5

👀 See It

Each gold dot is a term of the sequence. As n grows, dots approach the red dashed line (L = 1). That's 'convergence'.

ε-N Definition

📏 Meaning of the ε-Band

①Larger n shrinks the gold band (ε-band)

②No matter how tiny ε is, all terms past some N stay inside

③That's the rigorous definition of convergence

ε-N Definition

∀ε > 0, ∃N : n ≥ N ⇒ |a_n − L| < ε

For every ε > 0 there exists N such that n ≥ N ⇒ |aₙ − L| < ε

Convergent vs Divergent

Choose sequence0

🔍 Three Patterns

①Mode 0: 1 + 1/n → 1 (clusters together)

②Mode 1: (−1)ⁿ(1 + 1/n) — oscillates, diverges

③Mode 2: n/(n+1) → 1 (approaches from below)

Limit Laws

Arithmetic of Limits

lim(a_n ± b_n) = α ± β, lim(a_n · b_n) = α · β

Sum/diff/product of limits when both sequences converge

Quotient

lim a_nb_n = αβ (β ≠ 0)

Quotient of limits when β ≠ 0

Squeeze Theorem

a_n ≤ c_n ≤ b_n, lim a_n = lim b_n = L ⇒ lim c_n = L

If two sequences squeeze a third to the same limit, that third converges to it

Worked Examples

Example 1

Find lim_n→∞ 2n+3n-1.

Divide numerator and denominator by n.

2n+3n-1 = 2 + 3/n1 - 1/n

As n → ∞, 3/n → 0 and 1/n → 0.

2 + 01 - 0 = 2

▸ 2

For limits of rational expressions, dividing by the highest power in the denominator is the standard move.

Example 2

Find lim_n→∞ sin nn.

Since −1 ≤ sin n ≤ 1, divide all sides by the positive number n.

-1n ≤ sin nn ≤ 1n

Both outer limits are 0, so by the squeeze theorem the middle is 0.

lim_n→∞ (-1/n) = lim_n→∞ 1n = 0

▸ 0

For bounded sequences like sin or cos, squeeze with ±1/n and apply the squeeze theorem.

Wrap-up

Key Limits

lim_n→∞ 1n^p = 0 (p > 0), lim_n→∞ rⁿ = 0 (|r| < 1)

1/n^p → 0; rⁿ → 0 when |r| < 1

CSAT-style

What is lim_n→∞ (√(n²+3n) − n)?

①1

②32

③2

④3

⑤∞ (diverges)

▸ ② 32

This is an ∞ − ∞ form with a root, so multiply by the conjugate to rationalize.

√(n²+3n) − n = 3n√(n²+3n) + n

Divide numerator and denominator by n; the denominator tends to 2.

3√(1 + 3/n) + 1 → 32

🎯 Exam Points

①Definition: master the ε-N flow

②Use limit laws once each part converges

③Diverging sequences (oscillating) have no limit

④Squeeze theorem when two limits agree

⑤Memorize 1/n^p → 0 and r^n → 0 (|r| < 1)

Limit of a Geometric Sequence

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