{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-function-limit",
  "semantic_family_id": "sf-function-limit",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "去心實數論域的函數極限"
  },
  "summary": {
    "zh-Hant": "在a附近排除a本身，以ε-δ控制函數值；點值不屬於去心極限條件。"
  },
  "statement": {
    "zh-Hant": "對在a附近所有x≠a都有定義的實函數f，lim_(x→a)f(x)=L，指每個ε>0都有δ>0，使0<|x−a|<δ時 |f(x)−L|<ε。本輪用仿射式、可去因子商及左右仿射分支，左右有限極限必須相等才有兩側極限。"
  },
  "formula": {
    "tex": "L(f,a)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>L</mi><mo>(</mo><mi>f</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "L",
        "glyph": "L"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "f",
          "glyph": "f"
        },
        {
          "type": "symbol",
          "name": "a",
          "glyph": "a"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "195dbbc9d05ecd3e5b744127dd00becaa8f8c9306bf3e6333dee192e008b0580"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫函數採去心實數論域R∖{a}，不要求f(a)存在，不包含孤立點或其他未明示的接近論域。"
    },
    {
      "id": "a2",
      "zh-Hant": "仿射mx+b的極限為ma+b；(x²−a²)/(x−a)只在x≠a可消因子，極限為2a，不因此給原商在a的值。"
    },
    {
      "id": "a3",
      "zh-Hant": "正有理ε=p/q、δ=u/v的仿射均勻界允許 |m|u q≤p v；等號合法，因輸入距離嚴格小於δ。m=0時所有正δ合法。"
    }
  ],
  "symbols": [
    {
      "token": "L",
      "role": "punctured_real_limit",
      "role_zh": "去心實數極限運算",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "punctured_real_function",
      "role_zh": "在中心外有定義的指定函數",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "approach_point",
      "role_zh": "實數接近中心",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "函數的定義域與陪域。"
    },
    {
      "id": "mko-tends-to-relation",
      "title_zh": "趨近關係",
      "reason_zh": "接近點不等於直接代入。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "實數距離與ε-δ誤差。"
    },
    {
      "id": "mko-factorization",
      "title_zh": "因式分解",
      "reason_zh": "在非零分母前提下消去x−a。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "去心極限排除中心，因此原函數在中心未定義或改了中心值，都不直接改變這個極限。這和連續性需要中心點值不同。",
      "仿射界從 |f(x)−(ma+b)|=|m||x−a| 得到。輸入距離嚴格小於δ，故 |m|δ=ε仍可保證輸出嚴格小於ε；不要將數列尾界的嚴格比較原樣套過來。",
      "左右仿射分支各自有極限；兩側值不相等時兩側有限極限不存在。只否定此兩側極限，不把其中一側的合法極限抹掉，也不以有限測點替一般函數背書。"
    ],
    "common_misconception_zh": "把極限當f(a)、除零後補值，或把δ邊界等號誤判為不合法。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示仿射／可去商／兩側模型可精確核對；一般函數極限與型別橋未附形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 punctured nhds、左右鄰域與仿射誤差界；不以總定義除零API替代去心資格。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-function-limit.md 整理；2026-10-07 脈衝 R02（使用者手動接續 2026-10-08 工作序列；實際日期仍為 2026-10-07），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-07"
  }
}