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  "body": "\n## Definition\n\nLet $D$ be a subset of $\\R$ or of $\\C$ and let $x_0$ be a limit point of $D$, possibly $\\infty$.\nAn *asymptotic scale* (or *asymptotic sequence*) at $x_0$ is a sequence of functions $\\varphi_0, \\varphi_1, \\varphi_2, \\dots \\colon D \\to \\C$ such that:\n\n1. each $\\varphi_n$ has no zero in $D \\cap V$ for some punctured neighbourhood $V$ of $x_0$;\n2. for each $n \\ge 0$, $\\varphi_{n+1}(x) = o\\big(\\varphi_n(x)\\big)$ as $x \\to x_0$ in $D$, in [[little o notation|00T5EPFQB9]].\n\n## Remarks\n\nExample: $\\varphi_n(x) = (x - x_0)^n$ on $D = \\R$ at a finite $x_0$.\nThese have no zero for $x \\ne x_0$, and $\\varphi_{n+1}(x) / \\varphi_n(x) = x - x_0 \\to 0$.\n\nMixed scales such as $1, \\frac{\\log x}{x}, \\frac1x, \\frac{\\log x}{x^2}, \\frac{1}{x^2}, \\dots$ at $\\infty$ are also asymptotic scales.\n\nNon-example: $\\varphi_n(x) = x^{-n}$ is not an asymptotic scale at $x_0 = 0$, since $\\varphi_{n+1}(x)/\\varphi_n(x) = 1/x$ does not tend to $0$ there; it is a scale at $\\infty$.\n\nNon-example: $\\varphi_n(x) = \\sin(x)\\, x^{-n}$ at $\\infty$ on $D = (1, \\infty)$ fails clause 1, since every $\\varphi_n$ has zeros at $x = k\\pi$ for all large $k$.\n\nSome authors drop clause 1 and write the condition 2 as $\\varphi_{n+1} = o(\\varphi_n)$ in the sense of an inequality $\\abs{\\varphi_{n+1}} \\le \\varepsilon \\abs{\\varphi_n}$ near $x_0$.\n",
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