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Calculus · Limits

Numerical Limits & Local Behavior Calculator

Inspect left- and right-hand numerical behavior for representative indeterminate limits using progressively smaller distances from the approach point.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Two-sided numerical evidence near a finite approach point

Left behavior

x→a⁻

Right behavior

x→a⁺

Two-sided limit

lim x→a f(x)=L when both sides approach L
Calculus · Limits

Inspect a two-sided limit numerically

The table samples both sides at shrinking distances. Numerical agreement supports a limit but is not a symbolic proof.

Inputs and displayed decimal results use at most three decimal places. Full internal precision is retained during calculation.

lim (x²−a²)/(x−a), as x→a

Choose a representative indeterminate form and inspect its local behavior.

Using the model

Symbols, assumptions and limitations

Interpretation notes

This calculator provides numerical evidence for a controlled set of standard limits. Agreement of sampled values is educational evidence, not a symbolic proof, and does not imply that the original expression is defined at the approach point.

Worked examples

See the method in practice

01

Removable discontinuity

For (x²−4)/(x−2), values from both sides approach 4 even though direct substitution gives 0/0.

02

Trigonometric limit

For sin x/x as x approaches 0 in radians, left and right samples both approach 1.

Questions

Frequently asked

Does numerical agreement prove a limit?

No. It supports a conjecture; a proof uses algebra, inequalities, or established limit theorems.

Why sample both sides?

A finite two-sided limit exists only when the left- and right-hand limits agree.

Can the function be undefined at the approach point?

Yes. A limit describes nearby behavior and may exist across a removable discontinuity.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Calculus Volume 1 — The Limit of a Function

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology