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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{R}$, let $f \\colon D \\to \\mathbb{R}$, let $a$ be a limit point of $D$, and let $L$ be a real number.\nThen $\\lim_{x \\to a} f(x) = L$ if and only if the following holds.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $|f(x) - L| \u003c \\varepsilon$ for all $x \\in D$ with $0 \u003c |x - a| \u003c \\delta$.\n\n## Remarks\n\nEach open interval around a point contains an interval $(L - \\varepsilon, L + \\varepsilon)$, and each such interval is open. This gives the equivalence with the [[definition|0TXTSEMHR5]].\nMany texts take this condition as the definition of the limit.\nMathlib states a form for metric spaces as `Metric.tendsto_nhds_nhds`.\n\n## Explanations\n\n### L1 human\n\nFor each error tolerance $\\varepsilon$, we can force $f(x)$ within $\\varepsilon$ of $L$.\nWe do this by taking $x$ within a suitable distance $\\delta$ of $a$.\n",
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