Limits of Sequences and Functions, Continuity
MathematicsFunctions & GraphsAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Functions & Graphs
Sequence mode: the points (n, uₙ) approach the line y = L; a draggable band L ± ε shows that from some index N₀ onward every term stays inside the band. Function mode: drag x toward x₀ from both sides to watch f(x) approach the left- and right-hand limits, spot a discontinuity or a point where the function is undefined at x₀, and draw a conclusion about continuity. Techniques mode works through indeterminate forms step by step: factorising or rationalising to remove a 0/0 hole, the squeeze theorem with x² sin(1/x) and sin x / x, l'Hôpital's rule for 0/0 and ∞/∞, and limits at infinity, each linked to the graph and a value table.
Lesson: Limits of sequences, limits of functions, continuity
What it shows
A sequence uₙ has limit L if, for every band L ± ε however narrow, all terms from some index N₀ onward lie inside it; (−1)ⁿ fits in no narrow band. A function has a limit at x₀ when its one-sided limits agree, and it is continuous there when that limit equals f(x₀). Techniques mode turns 0/0 and ∞/∞ forms into exact values: factorising or multiplying by the conjugate removes a hole, the squeeze theorem traps x² sin(1/x) and sin x / x between simpler functions, l'Hôpital's rule compares f/g with f′/g′, and dividing by the highest power gives limits at infinity.
How to use
In Sequence, pick a sequence and drag vertically on the chart to narrow ε; predict how N₀ changes. In Function, set x₀, drag the red point towards it from each side or press Run, and compare the one-sided limits with f(x₀). In Techniques, choose a case and press Next step to reveal the working one step at a time, while you drag the red point and watch the value table.
Parameters you can change
- Object Sequence uₙ, Function f(x), Techniques for evaluating limits
- Sequence uₙ = 1/n, uₙ = (−1)ⁿ/n, uₙ = (2n + 1)/(n + 1), uₙ = (−1)ⁿ (no limit)
- ε (half-width of the band around L) 0.02–0.5
- Function f(x) = (x² − 1)/(x − 1), f(x) = sin x / x, f(x) = |x|/x, f(x) = x² (continuous)
- Point x₀ where the limit is examined -3–3
- Technique (Techniques mode) Factorise: (x² − x − 6)/(x − 3), x → 3, Conjugate: (√(x + 4) − 2)/x, x → 0, Squeeze: x² sin(1/x), x → 0, Squeeze: sin x / x, x → 0, L'Hôpital 0/0: ln x / (x − 1), x → 1, L'Hôpital ∞/∞: ln x / x, x → ∞, At infinity: (2x² − 3x + 1)/(x² + 4)
Questions to explore
- For ε = 0.1, from which index N₀ onward do all terms of uₙ = 1/n lie inside the band?
- Why does multiplying by √(x + 4) + 2 remove the 0/0 form of (√(x + 4) − 2)/x as x → 0?
- Which of the four functions are continuous at x₀ = 0, and what goes wrong for the others?
