Function limitAtPosInf

src/compute-engine/symbolic/limit.ts:206–304

(
  e0: Expression,
  x: string,
  ce: ComputeEngine,
  depth: number
)

// ──────────────────────────────────────────────────────────────────────────
// Finite point
// ──────────────────────────────────────────────────────────────────────────

/**
 * Sign of `c_v·(x − a)^v` (the leading Laurent term) as `x → a` from `dir`
 * (+1 right, −1 left, 0 two-sided), or `undefined` when indeterminate:
 * a two-sided approach with odd `|v|` (the two sides disagree), or a
 * leading coefficient whose sign the engine cannot decide.
 */
function laurentSignNear(
  L: { v: number; coeff: (p: number) => Expression },
  dir: number
): 1 | -1 | undefined {
  const odd = Math.abs(L.v) % 2 === 1;
  if (dir === 0 && odd) return undefined;
  const c = L.coeff(L.v).evaluate();
  if (!c.isValid || c.isNaN === true) return undefined;
  let s: 1 | -1;
  if (c.isPositive === true) s = 1;
  else if (c.isNegative === true) s = -1;
  else return undefined;
  // From the left, x − a = −t (t > 0): odd powers flip the sign.
  if (dir === -1 && odd) s = s === 1 ? -1 : 1;
  return s;
}

/**
 * Signed-infinity resolution of a pole (negative-valuation Laurent
 * expansion) — the 2026-07-10 convention decision: a *directional* limit at
 * a pole resolves to `±∞` from the sign of the leading coefficient and the
 * parity of the valuation; a *two-sided* limit resolves only when both
 * sides agree (even valuation, `lim 1/x² = +∞`). An odd-valuation two-sided
 * limit (`lim 1/x` at 0) stays inert — the engine does not produce
 * `ComplexInfinity` limits.
 */
function signedPoleInfinity(
  L: { v: number; coeff: (p: number) => Expression },
  dir: number,
  ce: ComputeEngine
): Expression | undefined {
  if (L.v >= 0) return undefined;
  const s = laurentSignNear(L, dir);
  if (s === undefined) return undefined;
  return s > 0 ? ce.PositiveInfinity : ce.NegativeInfinity;
}

/**
 * Operator heads with JUMP discontinuities: piecewise-constant-like
 * functions whose value changes discontinuously as their argument crosses a
 * threshold. Direct substitution is unsound for a subterm of one of these
 * sitting ON its jump: the value AT the point is not the limit from either
 * side (`sgn(0) = 0`, but `lim_{x→0⁻} sgn x = −1`). The finite-point
 * strategies resolve such a subterm per DIRECTION instead — see
 * `substituteAtFinitePoint`. Kink-continuous heads (`Abs`) do not belong
 * here: their value at the point IS the two-sided limit.
 */
const JUMP_FNS = ['Sign', 'Heaviside', 'Floor', 'Ceil', 'Round', 'Mod'];