Asymptotic scale

Remarks

Example: $\varphi_n(x) = (x - x_0)^n$ on $D = \R$ at a finite $x_0$. These have no zero for $x \ne x_0$, and $\varphi_{n+1}(x) / \varphi_n(x) = x - x_0 \to 0$.

Mixed 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.

Non-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 $\infty$.

Non-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 $k$.

Some 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 $x_0$.