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  "aliases": [
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  "body": "\n## Statement\n\nLet $I$ be an open interval and let $c \\in I$.\nLet $f, g$ be real functions differentiable on $I \\setminus \\{c\\}$, with $g'(x) \\ne 0$ for all $x \\in I \\setminus \\{c\\}$.\nAssume that $f(x) \\to 0$ and $g(x) \\to 0$ as $x \\to c$, and that $f'(x)/g'(x) \\to L$ as $x \\to c$.\nThen $f(x)/g(x) \\to L$ as $x \\to c$.\n\n## Remarks\n\nThe proof uses the [[Cauchy mean value theorem|02SYZAWM2K]] on intervals between $c$ and $x$.\nSimilar forms hold for the [[indeterminate form|0S7G92KKA7]] $\\infty/\\infty$, for one-sided limits, and for $x \\to \\pm\\infty$.\nFor example, $\\lim_{x \\to 0} \\frac{1 - \\cos x}{x^2} = \\lim_{x \\to 0} \\frac{\\sin x}{2x} = \\frac{1}{2}$.\nMathlib states a form as `deriv.lhopital_zero_nhds`.\n\n## History\n\nJohann Bernoulli found the rule and sent it to Guillaume de l'Hôpital in a letter of 1694.\nL'Hôpital published it in his textbook of 1696.\n\n## Explanations\n\n### L1 human\n\nNear $c$, both $f$ and $g$ are close to their tangent lines through $(c, 0)$.\nSo their ratio is close to the ratio of their slopes.\n",
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    "name": "L'Hôpital's rule",
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  "name": "L'Hôpital's rule",
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