Limit Calculator mechanism

Model scope

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.

• 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.

Units and conventions

• 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.

Inspectable vector

One of 4 source-bound review vectors:

{
  "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."
}

Failure boundary

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.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json