HomeCalculatorsMathNumerical CalculusTwo-Sided Limit Calculator
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Two-Sided Limit Calculator

Numeric two-sided limit of a polynomial, sin, or exp at a finite x. Same engine as Numerical Derivative. Not one-sided. Not L’Hôpital. Not a second engine. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published methods
Assurance
Core
Declared partition coverage
PASS · 7/7 declared partitions (tiny-step, subnormal-exp, central, forward, backward, limit, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Numeric two-sided limit of poly, sin, or exp at a finite x, with an engine-linked figure of f around x.
Scope
The Result Card includes an engine-linked figure of f around x (horizontal at the two-sided probe). Same engine as /numerical-derivative. Not symbolic CAS lim. Not one-sided. Not L’Hôpital.
Verification
Engine tested · Source checked · v1.1.2 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published methods· View Manifest · CVP overview · Specification
Versions
Calculation 1.1.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-07.derivative-elementary-stability
Verification revision
2026-10-07.derivative-elementary-stability · 1/1 property · digest 27f3f8a25a41
Legacy regression
14/14 tests · Production surface contract 4/4
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 3/3 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
28/28 golden · 2/2 CVP boundary · 8/8 invalid · 1/1 property · 11/11 cross-interface · 1/1 cross-calculator · 4/4 CVP contract · Manifest
Sources
Sources
Evidence
4 legacy golden · 3 legacy boundary · legacy regression suite · 28/28 oracle-backed golden · 8/8 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.1.2 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Two-sided probe[f(x+h) + f(x−h)] / 2
Exact (continuous f)lim_{t→x} f(t) = f(x)
iSame engine as /numerical-derivative. Not one-sided left/right. Not x→∞. Not L’Hôpital. Not symbolic CAS. Default h = 10⁻⁶. Degree ≤ 8.

How to use

1

Choose f and x

Poly needs coeffs (constant first). sin and exp are f(t)=sin t and eᵗ. Finite x only.

2

Read the two-sided value

Left and right probes are shown. Finite-difference derivatives stay on /numerical-derivative.

Example calculations

Common configurations with formula and result.

ϟ

x² at 3

coeffs 0,0,1

lim = 9
9
ϟ

sin at 0

two-sided

sin 0
0

Two-Sided Limit calculator specification

Version 1.1.2 · Engine tested

Calculation status

Review policy · Evidence

Definition
lim_{t→x} f(t) is estimated by [f(x+h)+f(x−h)]/2. This page is method=limit on math.numerical.derivative. Polynomial, sin, and exp are continuous, so the exact check is f(x). Not a second engine. Not one-sided. Not L’Hôpital. Not a CAS.
What it calculates
Numeric two-sided limit of poly, sin, or exp at a finite x, with an engine-linked figure of f around x.
Inputs
  • integrand
  • x
  • h?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
  • left
  • right
  • gap
Formula
[f(x+h) + f(x−h)] / 2
Assumptions
  • The Result Card includes an engine-linked figure of f around x (horizontal at the two-sided probe). Same engine as /numerical-derivative. Not symbolic CAS lim. Not one-sided. Not L’Hôpital.
  • Not one-sided left/right
  • Not L’Hôpital
  • Not x→∞
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • exp(x) underflow to zero → NUMERICAL_RANGE_ERROR
  • Non-finite function probes, derivative or error → NUMERICAL_RANGE_ERROR
  • h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • h > 1 → VALUE_OUT_OF_RANGE
  • unknown method (including left, right, lhopital, autodiff, cas) → INVALID_MODE
Example
method=limit integrand=poly coeffs=0,0,1 x=3 → 9
Validation cases

2 published on this page · 14/14 tests · Production surface contract 4/4 · View evidence

  • method=limit integrand=poly coeffs=0,0,1 x=3 → exact=9
  • method=left integrand=sin x=0 → INVALID_MODE
Sources
Calculation version
1.1.2

Background

Interpretation and common distinctions.

What does this two-sided limit calculator compute?

lim_{t→x} f(t) is estimated by [f(x+h)+f(x−h)]/2.

The published formula or method is [f(x+h) + f(x−h)] / 2. The following references support the stated mathematics or method:

  • Gilbert Strang, Calculus — Chapter 2: Derivatives (MIT OpenCourseWare): Two-sided and one-sided limits, continuity, and functions whose limits do not exist. Finite numerical samples alone are not proof of a mathematical limit; CalculatorX’s sampling limitations are stated separately. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

A finite step gives a numerical estimate. Very small steps can amplify roundoff, while large steps introduce truncation error; finite samples alone do not prove an arbitrary limit.

The worked example uses coeffs 0,0,1 with lim = 9. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.numerical.derivative v1.1.2 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Other calculators in this family: Brent Method Calculator, Definite Integral Calculator, Euler Method Calculator, Gauss Quadrature Calculator, Golden Section Search Calculator, Nonlinear Equation Calculator, Numerical Derivative Calculator, Numerical Interpolation Calculator . Explore all Numerical Calculus.

→ Brent Method Calculator Brent’s method (golden section plus inverse parabolic interpolation) for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not root-finding Brent. Runs locally. › → Definite Integral Calculator Composite trapezoid, Simpson 1/3, or Gauss–Legendre of a polynomial, sin, or exp on [a, b]. Not indefinite symbolic integrate. Runs locally. › → Euler Method Calculator Forward Euler for a scalar first-order IVP y′=f(t,y). Same engine as Ordinary Differential Equation. Not RK4, not rk45, not a system. Runs locally. › → Gauss Quadrature Calculator n-point Gauss–Legendre quadrature of a polynomial, sin, or exp on [a, b]. Same engine as Definite Integral. Not indefinite CAS. Runs locally. › → Golden Section Search Calculator Golden-section search for a minimum of a polynomial, sin, or exp on [a, b]. Same engine as One-Dimensional Optimization. Not Brent, not multivariate. Runs locally. › → Nonlinear Equation Calculator Find one approximate real zero of a polynomial or sin x with Newton, bisection, or secant. Uses the Numerical Root engine. Runs locally. › → Numerical Derivative Calculator Finite-difference derivative of a polynomial, sin, or exp, plus a numeric two-sided limit. Central, forward, backward, or limit. Discovery /two-sided-limit. Not symbolic d/dx. Runs locally. › → Numerical Interpolation Calculator Piecewise-linear, Lagrange, or natural cubic spline of tabulated (x, y). Not regression, not CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second derivative engine?

No. It is method=limit on math.numerical.derivative, the same engine as /calc/math/numerical-derivative. Forward / backward / central differences stay on that page.

Do you ship one-sided limits or L’Hôpital?

No. method=left, method=right, and method=lhopital are INVALID_MODE. This page is the two-sided numeric probe only. Not a CAS.

Where does this run?

Locally in the browser by default. REST and MCP call the same derivative engine.