Limit Calculator
The limit, approached honestly from both sides: the h-ladder printed, the verdict as the two sides’ average with the gap named — and the hole case taught live, where f(a) refuses but the limit sails through.
Limit Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Numeric approach to h = 1e-6 — not an algebraic proof
- One point per run; no one-sided-only mode (both sides always shown)
In short: lim x→3 of x²: the sides walk in with h from 0.1 down to 1e-6 and end at 8.999994 (left) and 9.000006 (right) — gap 0.000012, well under the agreement threshold, so the limit prints 9.000000 and f(3) = 9 agrees: continuous. The hole chip is the lesson: (x² − 1)/(x − 1) at x = 1 refuses f(1) outright (division by zero) while both sides slide to 2 — the point never mattered, only the approach.
Formula
walk x = a − h and a + h as h: 0.1 → 1e-6
sides agree (gap under the threshold) → the limit is their average
a limit is about the ROAD into the point, never the point itself.
Worked Example
- Walk both sides. evaluate f(a − h) and f(a + h) for h = 0.1 down to 1e-6 — the ladder is printed so the convergence is visible, not asserted.
- Verdict. if the final sides sit within the agreement threshold, the limit is their average. If they disagree — or run off in opposite directions — the honest verdict is does-not-exist.
- Check the point. f(a) itself is evaluated and named: equal to the limit means continuous; refused while the limit survives means a hole — the textbook’s favorite distinction, live.
Three endings exist and the page prints which one you got: AGREE (the limit is the average — x² at 3 gives 9.000000), HOLE (f(a) refused, limit fine — the (x² − 1)/(x − 1) chip), and DISAGREE (the sides part ways — 1/x at 0 runs off to −huge and +huge, no limit). All three are answers; none is an error.
Strengths & Limits Of This Model
Where this engine is strong
- The ladder is printed — convergence is shown, not asserted
- The hole case is handled as a lesson instead of an error
Where it stops
- No symbolic limit laws (L'Hôpital and friends)
- No limits at infinity
Practical Use Cases
Continuity checks
does the curve actually pass through the point?
Derivative prep
the difference quotient IS a limit — see the Derivative page
Behavior hunting
what happens near the scary values of a function
Methodology & Editorial Standards
Expression parsed with the site grammar (radians, variable x). f evaluated at a − h and a + h for h = 0.1 down to 1e-6; the ladder prints each pair. Final sides within the gap threshold produce the limit as their average; a wide gap, or sides of opposing huge magnitude, produce the does-not-exist verdict with the final values shown. f(a) itself is evaluated separately and its verdict (value, refusal) printed on the point card — a refusal there never blocks the limit.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Limit Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does a limit actually ask?
One question: where is f HEADING as x closes in on a — regardless of whether f(a) exists when you get there. The page walks in from both sides at h = 0.1 down to 1e-6 and shows you every step, because a limit claimed without the approach is a guess in notation costume.
Why can f(a) be undefined while the limit exists?
Because the limit only samples NEAR the point, never AT it. The hole chip is the canonical case: (x² − 1)/(x − 1) factorizes to (x + 1)(x − 1)/(x − 1) — at every x except exactly 1 it behaves like x + 1, so the sides glide to 2 while f(1) sits on a 0/0 refusal. The graph has a pinprick hole at (1, 2) and the limit walks straight across it.
What makes the two sides DISAGREE?
The function does different things on the way in from the left versus the right. 1/x at 0 is the extreme case: from the left it dives toward negative unbounded, from the right it climbs toward positive unbounded — no single destination exists, so the verdict is does-not-exist, printed with the two final values that prove it.
Why print the whole h-ladder?
Because convergence is the evidence. Watching 9.1, 9.01, 9.001, … march toward 9 from both sides is qualitatively different from being told “limit = 9”: the ladder shows the function settling, and it also EXPOSES trouble — oscillation or drift — that a single final number would hide.
How does the page decide the sides “agree”?
The final left and right values (h = 1e-6) must sit within a small gap of each other; the printed limit is their average, and the gap is printed beside the verdict. On x² at 3 the final sides are 8.999994 and 9.000006 — gap 0.000012 — and the average is exactly 9. The threshold is honesty about measurement, not a claim of algebraic proof.
What is continuity, in this page’s terms?
Three-way agreement: the left limit, the right limit, and f(a) all the same. The point card prints all three verdicts — x² at 3 is continuous (all 9); the hole chip is NOT continuous (limit 2, f(1) refused) even though the limit exists. Continuity is a stronger condition than having a limit, and the page keeps them visibly separate.
Why does 1/x refuse rather than print a huge number?
Because “limit = ∞” is not a value — infinity is a DIRECTION of travel, not a destination, and the two sides here travel in OPPOSITE directions anyway. Printing a gigantic number would invite dividing by it later. The DNE verdict with both final values is the honest output.
What does the page do at a jump (step function)?
The sides disagree by the height of the jump and the verdict says so. Drive a piecewise-ish expression like abs(x)/x at 0: left side −1, right side +1, gap 2 — the page reports does-not-exist with both one-sided values visible, which is exactly what a calculus course wants you to write.
Is the limit always reached by six steps of h?
For the smooth cases, yes comfortably. For functions with violent fine structure the ladder may still be moving at h = 1e-6 — and the printed gap then confesses it: a wide gap is the page saying “the road in is rough, treat this verdict with care.” The ladder card exists precisely so you can tell a settled limit from a struggle.