Limit Calculator
Where a function is heading, approached from both sides, with l'Hôpital's rule applied when the form is indeterminate.
Results
What this tool does
A limit asks where a function is going, not where it is — which matters most exactly where the function is not defined at all. This page approaches your point from the left and from the right in shrinking steps and shows the table, so you can see the convergence rather than just being told a number. When the expression is a quotient that evaluates to zero over zero, it also applies l'Hôpital's rule using exact symbolic derivatives, and tells you it did. If the two sides disagree, it says the limit does not exist rather than picking one.
Formula
lim f(x), x → a⁻ e x → a⁺ · 0÷0 ⟹ lim f ÷ g = lim f′ ÷ g′ (l'Hôpital)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
expr | The expression, in x | — |
towards | x goes to | — |
point | Which point | — |
LM | The limit | — |
VD | What happens there | — |
LE | Coming from the left | — |
LD | Coming from the right | — |
TB | Step · value from the left · value from the right | — |
LH | Was l'Hôpital's rule applied? | — |
LX | The quotient of the derivatives | — |
LV | What that quotient gives | — |
IN | How the page read it | — |
Worked example
- The expression, in xsin(x)/x
- x goes topoint
- Which point0
- The limit1.0000000000
- What happens thereBoth sides agree — the limit exists
- Coming from the left1.0000000000
- Coming from the right1.0000000000
- Step · value from the left · value from the right 0.1 0.99833417 0.99833417 0.01 0.99998333 0.99998333 0.001 0.99999983 0.99999983 0.0001 1 1 0.00001 1 1 0.000001 1 1
- Was l'Hôpital's rule applied?Yes
- The quotient of the derivativescos(x)
- What that quotient gives1.0000000000
- How the page read itsin(x) / x
Limitations
- The calculation runs entirely in your browser. The values you type are never sent to a server.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
Frequently asked questions
Is a limit found this way a proof?
No, and it is worth being clear about that. The page approaches the point from both sides with steps of a tenth, a hundredth and so on down to a millionth, and reports what it sees. That is strong evidence and it is how most people first check an answer, but it is not a proof — a function can behave one way at every point you sample and differently in between. Where the expression is a quotient that gives 0/0, the page also applies l'Hôpital's rule using the exact symbolic derivatives, and that part is a genuine derivation. It says which of the two it did.
What does "the two sides disagree" mean?
That there is no limit. A limit exists only when approaching from the left and approaching from the right give the same answer; 1/x at zero heads to minus infinity from one side and plus infinity from the other, so the limit simply does not exist, even though both one-sided limits do. The page shows both numbers so you can see why, and refuses to print a single value in that case — showing one side as though it were the answer would be worse than showing nothing.