{
  "aliases": [
    "two-path-test"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0C0KPAKK9Y"
    }
  ],
  "body": "\n## Summary\n\nLet $f$ be a function on $D \\subseteq \\mathbb{R}^n$ and let $a$ be a limit point of $D$.\nLet $\\gamma \\colon (0, 1] \\to D$ be a path, for example a [[curve|0G7WM8DX4P]], with $\\gamma(t) \\ne a$ for each $t$ and $\\gamma(t) \\to a$ as $t \\to 0^+$.\nIf $\\lim_{x \\to a} f(x) = L$, then $\\lim_{t \\to 0^+} f(\\gamma(t)) = L$.\nSo if two such paths give different limits, or one path gives no limit, then the [[limit|0J8Y1NCY0K]] of $f$ at $a$ does not exist.\n\n## Remarks\n\nFor $f(x, y) = xy / (x^2 + y^2)$ on $\\mathbb{R}^2 \\setminus \\{(0, 0)\\}$, the path $y = m x$ gives the limit $m / (1 + m^2)$ at the origin. This depends on $m$, so $f$ has no limit at $(0, 0)$.\nEqual limits along all straight lines do not prove that the limit exists. For $f(x, y) = x^2 y / (x^4 + y^2)$, each line through the origin gives the limit $0$, but the parabola $y = x^2$ gives $1/2$.\n\n## Explanations\n\n### L1 human\n\nIn one variable you can approach a point only from the left or from the right.\nIn the plane you can approach a point along infinitely many routes: lines, parabolas, spirals.\nA limit must give the same answer for every route, so one route that disagrees is enough to show that there is no limit.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "two-path-test"
    ],
    "kind": "concept",
    "msc": [
      "26B05"
    ],
    "name": "Limit along different paths",
    "sources": [
      {
        "locator": "Functions of more than one variable",
        "url": "https://en.wikipedia.org/wiki/Limit_of_a_function"
      }
    ],
    "tag": "0BJ6H85DRQ",
    "uses": [
      "0J8Y1NCY0K",
      "0G7WM8DX4P"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/0BJ6H85DRQ/",
  "links": [
    {
      "field": "uses",
      "target": "0J8Y1NCY0K"
    },
    {
      "field": "uses",
      "target": "0G7WM8DX4P"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0BJ6H85DRQ.md",
  "msc": [
    "26B05"
  ],
  "name": "Limit along different paths",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0BJ/0BJ6H85DRQ.md",
  "status": null,
  "tag": "0BJ6H85DRQ",
  "uri": "m2://0BJ6H85DRQ",
  "url": "https://math2.tamnd.com/wiki/limit-along-different-paths/",
  "writing": {
    "tag": "0BJ6H85DRQ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}