Source: https://guthlabs.ai/docs/instruments/limit-calculator/
instrument / verified-clean
# Limit Calculator
The current ledger marks this instrument verified clean within the source-bound verification corpus.
Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**
## Inspectable records
[01**Overview**Purpose and lifecycle]()[02**Mechanism**Method and assumptions]()[03**Implementation**Binding and validation]()[04**Qualification**Evidence and errata]()
## Ledger state
The current ledger marks this instrument verified clean within the source-bound verification corpus.
**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)
**Tranche**: tranche-8
**Evidence ID**: t8-022
**Ledger rows**: 1
No ledger note is attached.
## Review findings
The supplied formulas contain standard ideas, but they are not a complete or reliable numerical limit algorithm. Exact symbolic simplification, domain analysis, adaptive precision, one-sided domain checks, and explicit uncertainty/DNE handling are required before the instrument can honestly claim to compute limits.
### Formula and scope findings
- A finite sequence of sampled points is not a sufficient standard algorithm for establishing a limit, especially for oscillatory, discontinuous, or rapidly varying functions.
- The L'Hôpital rule is stated without its required hypotheses: the original quotient must have a 0/0 or ∞/∞ form, the functions must be differentiable on a punctured neighborhood, the denominator derivative must be nonzero there, and the resulting derivative quotient must have an appropriate limit.
- Numerical differentiation for L'Hôpital can introduce step-size, cancellation, overflow, and noise errors; the specification does not define derivative estimation, precision, tolerance, retry logic, or certification criteria.
- The specification does not define a reliable method for distinguishing a removable discontinuity, a divergent infinite limit, and numerical overflow.
- One-sided limits require domain-aware sampling; the specification does not state how invalid function evaluations, domain boundaries, or approach directions are handled.
- The claimed support for ∞ and DNE is underspecified: no extended-real representation, divergence test, oscillation test, or confidence criterion is defined.
### Unit assumptions
- The independent variable and approach point must use the same units.
- The function output must have a consistent unit; the reported limit has that same output unit.
- For trigonometric, exponential, and logarithmic functions, arguments are assumed dimensionless unless the function definition explicitly supplies a scale.
- A one-sided direction is interpreted in the units of the independent variable; x→a+ means values greater than a and x→a− means values less than a.
## Boundary hardening
**BREAK_CLAIMED** · SELF_FLAGGED
The calculator is vulnerable at removable-discontinuity and branch-sensitive inputs: direct endpoint evaluation can report undefined instead of the correct limit, and a numeric approach can incorrectly collapse unequal one-sided limits into a single result. The infinity convention also requires explicit signed-direction handling.
## Errata and corrections
No applied erratum is attached to this unique-name record.
## Inspectable example
4 review vectors are recorded. One source-bound sample is shown below.
````
{
"inputs": {
"function": "(x^2 - 1)/(x - 1)",
"approach_point": "1",
"direction": "two-sided"
},
"expected_outputs": {
"canonical_exact_limit": "2",
"indeterminate_form_at_substitution": "0/0",
"limit": "2",
"spot_checks": [
{
"at": 0.999,
"input_side": 1.999,
"answer_side": 2
},
{
"at": 1.001,
"input_side": 2.001,
"answer_side": 2
}
]
},
"reference_data": {
"identity": "x^2 - 1 = (x - 1)(x + 1)",
"domain_restriction": "x != 1"
},
"working": "Substitution gives (1^2 - 1)/(1 - 1) = 0/0. Factor: (x^2 - 1)/(x - 1) = ((x - 1)(x + 1))/(x - 1) = x + 1 for x != 1. Therefore lim_{x->1} (x^2 - 1)/(x - 1) = lim_{x->1}(x + 1) = 1 + 1 = 2. Spot check: at x=0.9990000000, (x+1)=1.9990000000; at x=1.0010000000, (x+1)=2.0010000000. The original expression is undefined exactly at x=1, but its limit is 2."
}
````
## Related workflows
No source-defined multi-instrument workflow membership is recorded.
## Sources and evidence
These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.
- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`
- `evidence-tranche8/tranche8-results.json`
- `BOUNDARY_HAZARD_CATALOG.json`