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  "body": "\n## Definition\n\nLet $f \\colon (0, \\infty) \\to (0, \\infty)$ be a [[measurable function|YYNCJSR0PJ]].\nThe function $f$ is *regularly varying* (at infinity) if for every real $a \u003e 0$ the limit\n$$g(a) = \\lim_{x \\to \\infty} \\frac{f(ax)}{f(x)}$$\nexists and satisfies $0 \u003c g(a) \u003c \\infty$.\n\n## Remarks\n\nFor such $f$ there is a real number $\\rho$ with $g(a) = a^\\rho$ for all $a \u003e 0$; this is Karamata's characterization theorem.\nThe number $\\rho$ is the *index* of $f$.\nEquivalently, $f(x) = x^\\rho L(x)$ where $L$ is a [[slowly varying function|EHJ7D5CRZS]].\nThe regularly varying functions of index $0$ are exactly the slowly varying functions.\n\nExample: $f(x) = x^2 \\log(e + x)$ is regularly varying of index $2$.\nIndeed $f(ax)/f(x) = a^2 \\cdot \\log(e + ax)/\\log(e + x)$, and the second factor tends to $1$.\n\nNon-example: $f(x) = e^x$ is not regularly varying.\nFor $a = 2$ the ratio $f(2x)/f(x) = e^x$ tends to $\\infty$, so the limit is not finite.\n",
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