Limit Calculator
Evaluate limits with L’Hôpital’s rule and two independent probe sequences — which is how it catches the limits that look convergent and are not.
Two ways of getting closer
A limit must be the same along every approach. These are two unrelated sequences heading to the same point — if they disagree, there is no limit.
L'Hôpital's rule
Why closer and closer is not a proof
- Getting closer and closer is not a proof. sin(1/x) near zero is the standard demonstration: probe at 0.1, 0.01, 0.001 and the answers jump about, but probe at 1/π, 1/2π, 1/3π and every single one is exactly 0. A tidy limit of zero, entirely false, produced by a perfectly reasonable choice of points.
- So this probes along TWO unrelated sequences and only reports a limit when they agree. If a limit exists, every sequence approaching the point gives the same answer — two that disagree are a proof that it does not exist, which is stronger than a picture of numbers settling down.
- L’Hôpital’s rule applies ONLY to 0/0 and infinity over infinity. Using it anywhere else is a confident error: the limit of (x+1)/(x+2) at 0 is 1/2, but differentiating top and bottom gives 1/1 = 1. The form is checked before the rule is applied, and it is checked again between rounds.
- The rule uses exact symbolic derivatives, so where it applies the answer is proved rather than estimated. That is a genuinely different kind of result from a column of numbers that look like they are settling.
- A one-sided limit can exist where the two-sided one does not. |x|/x is -1 from the left and +1 from the right, so each side has an answer and the point itself has none — reporting either one alone would be half the truth.
- Growing without bound is not a value. Saying a limit "is infinity" describes what the function does rather than naming a number it reaches, and this keeps those two cases apart.
How to use
- Type an expression in x and the point it approaches.
- Read the verdict, and the two probe sequences behind it.
- Check whether the rule applied, and what form justified it.
Frequently asked questions
Why does it probe twice?
Because a limit must be the same along EVERY approach, so one sequence proves nothing. sin(1/x) near zero is the standard trap: sample at points that land near multiples of pi and every answer is essentially zero, a tidy and completely false limit. Two unrelated sequences make that visible instead of hiding it.
Can probing be wrong even when both sequences agree?
Yes, and the tool shows a case. For (1 - cos x) over x squared, both sequences settle confidently on 0 while the true limit is one half — because near zero, 1 - cos(x) is the difference of two numbers that agree in every digit a computer holds, so it cancels to exactly zero. Agreement between sequences is evidence, never proof.
When does L’Hôpital’s rule apply?
Only to 0/0 and infinity over infinity. Using it elsewhere is a confident error: the limit of (x+1)/(x+2) at 0 is one half, but differentiating top and bottom gives 1/1 = 1. The form is checked before the rule is used and again between rounds, and the tool refuses rather than producing that 1.
Is the rule’s answer better than probing?
It is a different kind of answer. The rule uses exact symbolic derivatives, so where it applies the result is proved rather than estimated — which is why it wins whenever the two disagree, and why the page says outright that the probing was wrong rather than quietly showing the better number.
What does "no limit" mean here?
Either the values keep swinging over the same range however close you get — that is oscillation, and sin(1/x) does it — or the two sides disagree, as with the absolute value of x over x, which is -1 from the left and +1 from the right. In the second case each side has a limit of its own and the point has none.
Is infinity a limit?
No, and the tool keeps the two apart. Saying a limit "is infinity" describes what the function does rather than naming a number it reaches, so growing without bound is reported as exactly that instead of as a value.
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