{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0GR2APWTAC"
    }
  ],
  "body": "\n## Statement\n\nLet $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers with $b \\ne 0$.\nAssume that $a_n \\to a$ and $b_n \\to b$ as $n \\to \\infty$.\nThen there is a natural number $N$ such that $b_n \\ne 0$ for all $n \\ge N$.\nMoreover, the sequence $(a_n / b_n)_{n \\ge N}$ converges to $a / b$.\n\n## Remarks\n\nThe main step is $1/b_n \\to 1/b$. For large $n$ we have $|b_n| \u003e |b|/2$, and then $|1/b_n - 1/b| = |b - b_n| / (|b_n|\\,|b|) \u003c 2 |b_n - b| / |b|^2$.\nThe [[limit of a product|0734RSVF2V]] of $(a_n)$ and $(1/b_n)$ then gives the result.\nThe hypothesis $b \\ne 0$ is needed. For $a_n = 1$ and $b_n = 1/n$, the quotient is $n$, which has no real limit.\n\n## Explanations\n\n### L1 human\n\nThe limit of a quotient is the quotient of the limits, when the limit of the denominator is not zero.\nFor example, $(3 + 1/n)/(2 + 1/n) \\to 3/2$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "40A05"
    ],
    "name": "Limit of a quotient of sequences",
    "role": "proposition",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Limit_of_a_sequence"
      }
    ],
    "tag": "03S6GWWKEZ",
    "uses": [
      "0XKWSSY8P2",
      "0734RSVF2V"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/03S6GWWKEZ/",
  "links": [
    {
      "field": "uses",
      "target": "0XKWSSY8P2"
    },
    {
      "field": "uses",
      "target": "0734RSVF2V"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/03S6GWWKEZ.md",
  "msc": [
    "40A05"
  ],
  "name": "Limit of a quotient of sequences",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/03S/03S6GWWKEZ.md",
  "status": null,
  "tag": "03S6GWWKEZ",
  "uri": "m2://03S6GWWKEZ",
  "url": "https://math2.tamnd.com/wiki/limit-of-a-quotient-of-sequences/",
  "writing": {
    "tag": "03S6GWWKEZ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}