Limit Calculator - Two-Sided Numeric Limits

How Limits Are Evaluated Numerically

A limit asks: what value does f(x) get close to as x gets close to a — without ever needing to equal a? This tool builds an approach table: it evaluates the function at points like a ± 0.1, a ± 0.01, a ± 0.001, and watches where the outputs settle. If the left and right sides settle on the same number, that is (almost certainly) the limit.

The Classic 0/0 Case

Try (x²−1)/(x−1) as x → 1. Direct substitution gives 0/0, but the table shows values homing in on 2 — the hole in the graph is “filled” by the limit. Factoring the numerator first, (x−1)(x+1)/(x−1) = x+1, confirms it exactly.

When Limits Fail

Jump: left and right approach different numbers — the two-sided limit does not exist.
Blow-up: values grow without bound near a vertical asymptote — the limit is ±∞ (or does not exist as a finite number).
Oscillation: sin(1/x) near 0 never settles — no limit.

Limits are the foundation of derivatives — see our derivative calculator.

Frequently Asked Questions

What is a limit in calculus?

A limit is the value a function approaches as the input approaches some point. It describes behavior near the point, even if the function is undefined exactly there.

What does 0/0 mean in a limit?

It is an indeterminate form — the limit could be anything. Try factoring, rationalizing, or L'Hôpital's rule; numerically, an approach table usually reveals the answer.

When does a limit not exist?

When the left and right sides approach different values, when the function oscillates forever, or when it grows without bound near a vertical asymptote.

Is this limit calculator free?

Yes — every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.