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  "body": "\n## Definition\n\nLet $L \\colon (0, \\infty) \\to (0, \\infty)$ be a [[measurable function|YYNCJSR0PJ]].\nThe function $L$ is *slowly varying* (at infinity) if for every real $a \u003e 0$\n$$\\lim_{x \\to \\infty} \\frac{L(ax)}{L(x)} = 1.$$\n\n## Remarks\n\nExample: $L(x) = \\log(e + x)$ is slowly varying.\nFor fixed $a \u003e 0$ and large $x$, $\\log(e + ax) = \\log x + \\log a + o(1)$ and $\\log(e + x) = \\log x + o(1)$, so the ratio tends to $1$.\nEvery positive measurable function with a positive finite limit at infinity is slowly varying.\n\nNon-example: $L(x) = x^\\rho$ with $\\rho \\ne 0$ is not slowly varying, because $L(ax)/L(x) = a^\\rho \\ne 1$ for $a \\ne 1$.\n\nThe uniform convergence theorem says that the limit is uniform for $a$ in each compact subset of $(0, \\infty)$.\nIn the language of regular variation, slowly varying functions are the regularly varying functions of index $0$.\n",
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