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  "body": "\n## Definition\n\nLet $f$ and $g$ be real functions defined on an interval $[x_0, \\infty)$.\nThe functions are asymptotically equivalent as $x \\to \\infty$ if $f(x) - g(x) = o(g(x))$ as $x \\to \\infty$, in the sense of [[little o notation|04TSHB9H8D]].\nThe notation is $f(x) \\sim g(x)$.\n\n## Remarks\n\nIf $g(x) \\ne 0$ for large $x$, then $f \\sim g$ means that $f(x)/g(x) \\to 1$.\nThe relation $\\sim$ is an equivalence relation.\nFor example, $x^2 + x \\sim x^2$. [[Stirling's formula|0KC0F51T6A]] states $n! \\sim \\sqrt{2 \\pi n}\\, (n/e)^n$.\nMathlib defines it for filters, as `Asymptotics.IsEquivalent`.\n\n## Explanations\n\n### L1 human\n\nTwo functions are asymptotically equivalent if their ratio tends to $1$.\nTheir difference may still grow, but it is small compared with the functions.\n",
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