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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^n$ and let $f \\colon D \\to \\mathbb{R}^m$.\nLet $a \\in \\mathbb{R}^n$ be a limit point of $D$: each open ball about $a$ contains a point of $D$ other than $a$.\nLet $L \\in \\mathbb{R}^m$.\nWhen this holds, $f(x)$ tends to $L$ as $x$ tends to $a$, with the notation $\\lim_{x \\to a} f(x) = L$.\nFor each $\\varepsilon \u003e 0$ there is a $\\delta \u003e 0$ such that $\\|f(x) - L\\| \u003c \\varepsilon$ for each $x \\in D$ with $0 \u003c \\|x - a\\| \u003c \\delta$.\nHere $\\|\\cdot\\|$ is the Euclidean norm of [[Euclidean space|0XPR1E85W0]].\n\n## Remarks\n\nA limit is unique when it exists, because $a$ is a limit point of $D$.\nThe value $f(a)$, and even whether $a$ is in $D$, has no effect on the limit.\nA function into $\\mathbb{R}^m$ tends to $L$ if and only if each component $f_i$ tends to $L_i$.\nTo show that a limit does not exist, it is enough to find two [[paths with different limits|0BJ6H85DRQ]].\n\n## Explanations\n\n### L1 human\n\nThe definition says that you can make $f(x)$ as close to $L$ as you want. For this, take $x$ close enough to $a$ but not equal to $a$.\nIn the plane, \"close to $a$\" means inside a small disc about $a$, so $x$ can come from every direction at once.\n",
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