Enter an expression in n and explore end behavior. See sampled values and an estimated limit. Download a report for sharing and review.
These examples show common infinite-limit behaviors for sequences and functions sampled at large n.
| Expression f(n) | Expected behavior | Why it happens |
|---|---|---|
| (3*n^2 + 2*n + 1)/(n^2 - 5) | Converges to 3 | Same degree; ratio of leading coefficients. |
| ln(n)/n | Converges to 0 | Polynomial growth dominates logarithmic growth. |
| exp(n)/n^5 | Diverges to +∞ | Exponential growth outpaces any power of n. |
| (-1)^n | No single limit | Alternates between -1 and 1 forever. |
The infinite limit is written as: limn→∞ f(n). This calculator estimates it by evaluating f(n) at increasing n values and classifying the trend.
Infinite limits describe end behavior, helping you forecast trends when inputs become very large. In growth models, a stable limit suggests saturation, while divergence signals runaway behavior. For example, if f(n)=(3n²+2n+1)/(n²−5), values approach 3 as n increases, matching the leading-coefficient ratio. In optimization, limits guide tolerance choices.
The calculator evaluates f(n) at n=10, 50, 100, 500, 1000, 5000, and 10000. This spread captures early transitions and late stabilization. If the last three values are close relative to scale, the tool reports finite convergence. If magnitudes grow steadily, it flags likely divergence. When values become undefined, such as ln(n−100) near n=10, the notes warn you to adjust the expression or domain.
Rational functions often converge when numerator and denominator share degree. For ln(n)/n, sampled values shrink toward 0 because n grows faster than ln(n). When exp(n)/n⁵ is tested, outputs explode quickly, reflecting the dominance of exponential growth over any polynomial term. If f(n)=1+1/n, the table shows 1.1 at n=10 and 1.0001 at n=10000, demonstrating fast stabilization.
Divergence is identified when successive samples increase in magnitude and do not stabilize. Oscillation is suspected when recent finite samples change sign multiple times. A classic case is (−1)ⁿ, which alternates between −1 and 1, so no single limit exists even though values remain bounded. Another practical example is sin(n), which produces scattered values in [−1,1] without settling, so the trend remains nonconvergent.
Some sequences converge slowly, especially when f(n)=L+k/n. The calculator estimates L using two large samples: L≈(n₂f(n₂)−n₁f(n₁))/(n₂−n₁). This improves stability for cases like 2−5/n, where direct samples may still drift slightly. If the extrapolated estimate disagrees with the last sample by more than 0.01% of scale, the confidence stays medium to prevent overclaiming.
CSV export stores the expression, classification, confidence, and the full sample table for reproducibility. PDF export creates a concise one-page summary suitable for sharing. The Plotly chart provides an immediate visual check: flattening curves suggest convergence, while steep slopes suggest divergence. Pair numeric output with algebraic checks, such as dominant-term comparison, to confirm exact results when needed.
Enter an expression in terms of n using +, -, *, /, ^, and parentheses. Supported functions include sin, cos, tan, ln, log, exp, sqrt, and abs.
It provides a numeric classification and an estimate based on large-n sampling. For rigorous proofs, confirm with algebraic techniques such as factoring, dominant-term comparison, or known growth-rate rules.
Undefined values occur when the expression is outside its domain at sampled n, such as division by zero or ln of a nonpositive number. Adjust the expression or ensure it is valid for large n.
The tool checks recent sampled values for repeated sign changes and lack of stabilization. Expressions like (-1)^n or sin(n) typically fail to settle to one value.
Confidence summarizes how clearly the samples support a classification. High confidence usually means strong stabilization or clear divergence. Medium confidence appears when extrapolation helps but small drift remains.
CSV is best for spreadsheets, plotting, and reproducible checks. PDF is best for sharing a compact snapshot of the expression, classification, confidence, and sample evaluations with colleagues or students.
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.