What is a limit of a sequence?
A sequence is a list of numbers in order: a₁, a₂, a₃, … The small number n is the position of the term. For example aₙ = 1/n gives 1, ½, ⅓, ¼, …
We ask one question: as n gets very, very big, where do the terms go? If they get closer and closer to one fixed number L, then L is the limit. We write
lim (n→∞) aₙ = L, or aₙ → L.
The sign ∞ (infinity) is not a number. It only says "n keeps getting bigger".
Convergent and divergent sequences
A sequence that has a limit is convergent (it converges). A sequence that has no limit is divergent (it diverges).
- Converges: 1/n → 0, and (2n+1)/n → 2.
- Diverges to infinity: aₙ = n grows without end. We write aₙ → ∞.
- Diverges by jumping (oscillating): (−1)ⁿ keeps jumping between −1 and 1.
The exact idea (ε band). Pick any tiny width ε (say 0.01). Draw a band from L − ε to L + ε. If, from some term onward, every term stays inside the band, and this works for every ε you can pick, then aₙ → L. A thinner band only means you have to wait longer.
A convergent sequence has only one limit.
Limit laws
Suppose aₙ → A and bₙ → B (both are real numbers). Then:
- Sum: aₙ + bₙ → A + B
- Difference: aₙ − bₙ → A − B
- Constant multiple: c·aₙ → c·A
- Product: aₙ·bₙ → A·B
- Quotient: aₙ / bₙ → A / B, only if B ≠ 0
Trick for fractions. When the top and bottom both become huge, divide both by the highest power of n. Then pieces like 1/n and 1/n² go to 0, and you can read the answer.
Example: (3n² + n) / (n² + 4). Divide by n²: (3 + 1/n) / (1 + 4/n²) → 3/1 = 3.
Rule of thumb for (polynomial)/(polynomial): same top and bottom degree gives the ratio of the leading numbers; smaller top degree gives 0; bigger top degree diverges to ±∞.
Limits of geometric sequences rⁿ
A geometric sequence multiplies by the same number r every time: r, r², r³, … What happens to rⁿ?
- −1 < r < 1 (for example ½ or −0.3): the terms shrink to 0.
- r = 1: the terms are all 1, so the limit is 1.
- r > 1: the terms grow without end; diverges to ∞.
- r = −1: the terms jump between −1 and 1; diverges.
- r < −1: the size grows and the sign flips; diverges (oscillates, getting bigger).
So rⁿ converges exactly when −1 < r ≤ 1. Use this on a mix such as (3ⁿ + 2ⁿ) / 3ⁿ: divide by 3ⁿ to get 1 + (2/3)ⁿ → 1 + 0 = 1.
Try it
In the 3D: press the button aₙ = rⁿ and slide r. Predict first: what will the bars do at r = 0.9? At r = −0.5? At r = 1.2? Then check. Next, look at the green band and make ε smaller. Find which term n you must pass so that all later bars are inside.
At home: take a calculator. Type 1, then keep pressing "× 0.5" (or ÷ 2). Write 10 results. Where are they going? Now do the same with "× 2". Compare.
Key formulas and definitions
- aₙ → L means: for every ε > 0, all terms from some n onward lie between L − ε and L + ε
- lim (aₙ ± bₙ) = A ± B; lim (c·aₙ) = c·A; lim (aₙ·bₙ) = A·B; lim (aₙ/bₙ) = A/B (B ≠ 0)
- lim 1/n = 0, lim 1/nᵏ = 0 (k > 0), lim c = c
- lim rⁿ = 0 if |r| < 1; = 1 if r = 1; diverges if r > 1 or r ≤ −1
- Fractions in n: divide top and bottom by the highest power of n
Worked examples
1. Find lim 1/n².
n² grows as n grows, so 1/n² gets tiny. Limit = 0.
2. Find lim (3n + 2)/n.
Divide top and bottom by n: 3 + 2/n. As n grows, 2/n → 0. Limit = 3.
3. Find lim (4n² + 1)/(2n² − n).
Divide by n²: (4 + 1/n²)/(2 − 1/n). Top → 4, bottom → 2. Limit = 4/2 = 2.
4. Does (n² + 1)/(n + 5) converge?
Divide by n: (n + 1/n)/(1 + 5/n). The top grows without end, the bottom → 1. So the terms grow without end: it diverges to ∞.
5. Find lim (5 + 1/n)(2 − 3/n).
First factor → 5, second → 2. By the product law the limit is 5 × 2 = 10.
6. Find lim (3ⁿ + 2ⁿ)/(3ⁿ − 2ⁿ).
Divide by 3ⁿ: (1 + (2/3)ⁿ)/(1 − (2/3)ⁿ). Since |2/3| < 1, (2/3)ⁿ → 0. Limit = 1/1 = 1.
7. For aₙ = 1/n, how big must n be so that every later term is within 0.3 of 0?
1/n < 0.3 means n > 3.33. So from n = 4 onward all terms are inside the band. Check: 1/3 = 0.33 is out, 1/4 = 0.25 is in.
8. Find lim (√(n² + n) − n).
Multiply and divide by (√(n² + n) + n): the top becomes n, so we get n/(√(n²+n) + n). Divide by n: 1/(√(1 + 1/n) + 1) → 1/(1 + 1) = ½.
Common mistakes
- Saying "the terms never reach 0, so 1/n has no limit". A limit is where the terms head to, not a term itself.
- Using ∞ as an ordinary number, such as ∞ − ∞ = 0. Rewrite the expression first (divide by the highest power).
- Thinking a bounded sequence must converge. (−1)ⁿ stays between −1 and 1 but has no limit.
- Using the quotient law when the bottom limit is 0. Then the law does not apply; simplify first.