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  "body": "\n## Summary\n\nA positive function $f$ on $(0, \\infty)$ is regularly varying if $f(ax)/f(x)$ has a positive finite limit as $x \\to \\infty$ for every $a \u003e 0$.\nThe limit is then always a power $a^\\rho$, and $f(x) = x^\\rho L(x)$ with $L$ a [[slowly varying function|EHJ7D5CRZS]].\nSo regularly varying functions behave like powers up to a slowly varying factor.\nThey are the natural comparison functions in [[Karamata's Tauberian theorem|DBEGF38V9H]].\n\n## Explanations\n\n### L1 human\n\nA regularly varying function scales almost like a power of $x$ when $x$ is large.\nIf $f(x) = x^2 \\log x$, then doubling $x$ multiplies $f$ by about $4$, and the error in this \"about\" goes away as $x$ grows.\nThe number $\\rho$ in $a^\\rho$ is the index; here it is $2$.\nThe exponential function is not regularly varying, because doubling $x$ squares $e^x$, which is far more than any fixed factor.\n",
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