Classifying 0/0 Limits — Formal Specification — 0/0 Rate Law (4)

Assumptions near x0 (power models).

f(x) ~ c_f * |x - x0|^(p_f)
g(x) ~ c_g * |x - x0|^(p_g)     # finite nonzero c_f, c_g
  • For the 0/0 case: require p_f > 0 and p_g > 0 (both vanish at x0).
  • The rule below also classifies non-0/0 ratios when one of p_f, p_g is 0 (not the focus here).

Rule (magnitude + alignment).

a_div := tanh( atanh(a_f) - atanh(a_g) )
if p_f > p_g -> < 0 , a_div >
if p_f = p_g -> < c_f / c_g , a_div >
if p_f < p_g -> < s * inf , a_div >   where s := sign(c_f / c_g)

Safety (alignment clamp).

# clamp before any atanh
a := clamp(a, -1 + eps_a, +1 - eps_a)     # default eps_a = 1e-6

Conservativity (collapse).

phi(< m , a >) = m

This reproduces the classical dominant-term outcome (0, finite constant, or +/-inf).

Proposition (rate law, power models).
If f(x) ~ c_f * |x - x0|^(p_f), g(x) ~ c_g * |x - x0|^(p_g), with c_f != 0, c_g != 0 and p_f, p_g >= 0, then as x -> x0:

if p_f > p_g  ->  f/g -> 0
if p_f = p_g  ->  f/g -> c_f / c_g
if p_f < p_g  ->  |f/g| -> +inf with sign sign(c_f / c_g)

Proof (sketch).

f/g = (c_f / c_g) * |x - x0|^(p_f - p_g)
case p_f - p_g:
  > 0  ->  |x-x0|^(positive) -> 0
  = 0  ->  constant ratio -> c_f/c_g
  < 0  ->  |x-x0|^(negative) -> +inf ; sign from c_f/c_g
QED

Notes.

  • Sidedness. If the original functions include odd factors of (x - x0) (suppressed by |.| in the model), one-sided signs can differ; report with SIDED(L/R) when detected.
  • Degenerates. If a leading coefficient vanishes on estimation (c_f = 0 or c_g = 0), promote to the next nonzero term or mark NOFIT / MULTI per QA rules.
  • Alignment independence. The magnitude class is determined solely by the rate exponents (and sign via c_f/c_g); alignment only records approach/quality and composes via rapidities.

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