Explore limits across major sequence families. See convergence rules, dominant terms, graphs, and worked outputs. Practice confidently with examples, formulas, FAQs, and export options.
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| Example sequence | Model | Expected limit | Reason |
|---|---|---|---|
| (3n² + 4n + 1) / (6n² - 2) | Polynomial ratio | 0.5 | Equal degree, so use leading coefficients. |
| (2·2ⁿ) / (5·3ⁿ) | Exponential ratio | 0 | The denominator base is larger. |
| √(5n² + 1) / ∛(2n³ + 7) | Radical ratio | 5^(1/2) / 2^(1/3) | Effective powers both equal 1. |
| (n!) / 2ⁿ | Factorial ratio | ∞ | Factorials dominate exponentials. |
| an+1 = 0.5an + 2, a0 = 1 | Linear recursion | 4 | |r| < 1, so solve the fixed point. |
If an = P(n) / Q(n), compare the highest powers of n. Lower numerator degree gives 0. Equal degrees give leading coefficient ratio. Higher numerator degree gives ±∞.
If an = (A·rn + B) / (C·sn + D), compare dominant behaviors. Exponential growth beats constants and decays. Among competing exponentials, the larger base dominates.
For (a np + b)1/k / (c nq + d)1/m, compare effective powers p/k and q/m. Equal effective powers give a1/k / c1/m.
For A(n!)unp / [B(n!)vcnnq], compare factorial exponents first. If they tie, compare c. If c = 1, compare polynomial exponents.
For an+1 = r an + d, the fixed point is L = d / (1 - r). The sequence converges when |r| < 1.
It handles polynomial ratios, exponential ratios, radical ratios, factorial ratios, and first-order linear recursive sequences. Each model uses a matching dominant-term rule, so you can test several common sequence-limit patterns on one page.
The plot is meant for trend checking, not proof. Early terms help you see convergence, growth, decay, or oscillation. The exact result still comes from the algebraic limit rules shown in the reasoning section.
Yes. The recursive model can report no finite limit for oscillating or diverging cases. Other models can also return undefined or infinite outcomes when the denominator vanishes or the dominant term forces unbounded growth.
Positive bases avoid alternating-sign behavior that needs a separate analysis. This keeps the calculator stable and readable for most classroom and exam-style exponential ratio problems.
Effective power is the exponent after applying the root. For example, √(n²) behaves like n, so its effective power is 2/2 = 1. Comparing effective powers shows which side grows faster.
Factorials grow faster than any fixed-base exponential as n increases. That is why sequences like n! / 2ⁿ usually become unbounded, while 2ⁿ / n! tends to zero.
For an+1 = r an + d, assume the terms approach L. Then L = rL + d, so L = d / (1 - r). This works when |r| < 1.
Yes. After calculating, download a CSV file for spreadsheet use or export a PDF report for notes, homework support, and documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.