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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 Engine tested · Specification checked · 14/14 tests · Production surface contract 2/2 · v1.1.0
Input interpretation
Enter values to calculate.
Result
—
Model
Numeric two-sided limit of poly, sin, or exp at a finite x.
Scope
Not symbolic CAS lim
Verification
Engine tested · 14/14 tests · Production surface contract 2/2 · Specification checked · v1.1.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
4 golden · 3 boundary · 7 property · Production surface contract 2/2 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.1.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract
PASS

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 1.1.0 · 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.
Inputs
  • integrand
  • x
  • h?
  • coeffs?
Outputs
  • value
  • exact
  • abs_err
  • left
  • right
  • gap
Formula
[f(x+h) + f(x−h)] / 2
Assumptions
  • Not symbolic CAS lim
  • Not one-sided left/right
  • Not L’Hôpital
  • Not x→∞
  • Polynomial degree ≤ 8
Units
  • dimensionless
Boundary conditions
  • 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 2/2 · View evidence

  • method=limit integrand=poly coeffs=0,0,1 x=3 → exact=9
  • method=left integrand=sin x=0 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.1.0

Other calculators in this family: Bessel I Calculator, Bessel J Calculator, Bessel K Calculator, Bessel Y Calculator, Beta Function Calculator, Brent Method Calculator, Definite Integral Calculator, DFT Calculator . Explore all Numerical Calculus.

→ Bessel I Calculator I_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel J Calculator J_n(x) for integer n=0…8. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel K Calculator K_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Bessel Y Calculator Y_n(x) for integer n=0…8 and x>0. Same engine as Special Functions. Not W_{-1}. Not CAS. Runs locally. › → Beta Function Calculator Complete Euler B(x, y) = Γ(x)Γ(y)/Γ(x+y) for x>0, y>0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally. › → 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. › → DFT Calculator Forward DFT X[k]=Σ x[n] e^{−2πi kn/N}, N≤64, unnormalized. Same engine as Discrete Fourier Transform. Radix-2 FFT is on /fft. Not inverse, not Laplace. Runs locally. ›
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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.