Limit Calculator

How to use the Limit Calculator

1. Type f(x) — the function whose limit you want. Powers with ^, trig and logs by name.
2. Type the point — a number like 2, or inf / -inf for end behavior.
3. Pick a direction and press Enter. Read the verdict, then scroll the approach table: the raw numbers are the evidence.

Reading the approach table

The table samples h = 10⁻¹ down to 10⁻⁸. A converging limit shows the f(x) column stabilizing — each row agreeing with the last to more digits. A diverging limit shows magnitudes exploding (×10 each row for 1/x). An oscillating one, like sin(1/x), jumps around with no pattern of settling. Learning to read these three signatures is half of understanding limits.

When the estimate says DNE

DNE means "does not exist," and the message names the reason. Sides disagree: try the left and right buttons separately to see each one-sided limit — piecewise functions and |x|/x do this. Blows up: the limit is infinite, which some textbooks write as = ∞ and others as DNE; both mean unbounded. Oscillates: no amount of zooming settles it — that's a genuine non-existence, not a computing artifact.

How do I type my function?

Use x as the variable: sin(x)/x, (x^2-4)/(x-2), 1/x. Supported: + - * / ^, parentheses, sin cos tan, ln (natural log), log (base 10), sqrt, abs, and the constants e and pi. Multiplication can be implicit — 2x and x(x+1) both work.

How does the calculator estimate the limit?

It evaluates your function at points getting closer and closer to the target: h = 0.1, 0.01, all the way down to 0.00000001, from each side. If the values settle toward one number, that's the estimate; if they blow up, it reports +∞ or −∞; if they keep swinging, it reports DNE. The whole approach table is shown so you can see the evidence.

What does DNE mean?

DNE = does not exist. It happens for three reasons: the left and right sides approach different values (like |x|/x at 0), one side blows up to infinity, or the values oscillate forever without settling (like sin(1/x) at 0). The calculator tells you which reason it found.

Can it do limits at infinity?

Yes — type inf or -inf as the point. Then it marches x outward (10, 100, …, 100000000) instead of inward, using the same settle-or-diverge logic. Only the meaningful direction is sampled: at +∞ it approaches from the left.

What is the difference between left, right, and two-sided?

Left (x→a⁻) only samples values below a; right (x→a⁺) only above a; two-sided requires both sides to agree. For 1/x at 0, the right limit is +∞ and the left is −∞, so the two-sided limit is DNE.

Is a numerical estimate the same as a proof?

No — it's strong evidence, not a proof. Numerical estimates can be fooled by functions that misbehave only extremely close to the point. Always confirm with algebra (factoring, rationalizing, squeeze theorem) for homework and exams.

Is my input sent anywhere?

No. The parser and all the sampling run in your browser with JavaScript. Nothing is uploaded, stored, or tracked.