202. Scales of infinity

1. Between any two terms $f(x)$, $F(x)$ of the series we can insert a new term $\varphi (x)$ such that $\varphi (x)$ tends to $\mathrm{\infty }$ more slowly than $f(x)$ and more rapidly than $F(x)$. [Thus between $\sqrt{x}$ and $\sqrt[3]{x}$ we could insert $x^{5/12}$: between $\sqrt{\log x}$ and $\sqrt[3]{\log x}$ we could insert $(\log x{)}^{5/12}$. And, generally, $\varphi (x)=\sqrt{f(x)F(x)}$ satisfies the conditions stated.]

2. Find a function which tends to $\mathrm{\infty }$ more slowly than $\sqrt{x}$, but more rapidly than $x^{\alpha }$, where $\alpha$ is any rational number less than $1/2$. [$\sqrt{x}/(\log x)$ is such a function; or $\sqrt{x}/(\log x{)}^{\beta }$, where $\beta$ is any positive rational number.]

3. Find a function which tends to $\mathrm{\infty }$ more slowly than $\sqrt{x}$, but more rapidly than $\sqrt{x}/(\log x{)}^{\alpha }$, where $\alpha$ is any rational number. [The function $\sqrt{x}/(\log \log x)$ is such a function. It will be gathered from these examples that incompleteness is an inherent characteristic of the logarithmic scale of infinity.]

4. How does the function $$f(x)=\{x^{\alpha }(\log x{)}^{{\alpha }^{\prime }}(\log \log x{)}^{\alpha ”}\}/\{x^{\beta }(\log x{)}^{{\beta }^{\prime }}(\log \log x{)}^{\beta ”}\}$$ behave as $x$ tends to $\mathrm{\infty }$? [If $\alpha \ne \beta$ then the behaviour of $$f(x)=x^{\alpha -\beta }(\log x{)}^{{\alpha }^{\prime }-{\beta }^{\prime }}(\log \log x{)}^{\alpha ”-\beta ”}$$ is dominated by that of $x^{\alpha -\beta }$. If $\alpha =\beta$ then the power of $x$ disappears and the behaviour of $f(x)$ is dominated by that of $(\log x{)}^{{\alpha }^{\prime }-{\beta }^{\prime }}$, unless ${\alpha }^{\prime }={\beta }^{\prime }$, when it is dominated by that of $(\log \log x{)}^{\alpha ”-\beta ”}$. Thus $f(x)\to \mathrm{\infty }$ if $\alpha >\beta$, or $\alpha =\beta$, ${\alpha }^{\prime }>{\beta }^{\prime }$, or $\alpha =\beta$, ${\alpha }^{\prime }={\beta }^{\prime }$, $\alpha ”>\beta ”$, and $f(x)\to 0$ if $\alpha <\beta$, or $\alpha =\beta$, ${\alpha }^{\prime }<{\beta }^{\prime }$, or $\alpha =\beta$, ${\alpha }^{\prime }={\beta }^{\prime }$, $\alpha ”<\beta ”$.]

5. Arrange the functions $x/\sqrt{\log x}$, $x\sqrt{\log x}/\log \log x$, $x\log \log x/\sqrt{\log x}$, $(x\log \log \log x)/\sqrt{\log \log x}$ according to the rapidity with which they tend to infinity as $x\to \mathrm{\infty }$.

6. Arrange $$\log \log x/(x\log x),(\log x)/x,x\log \log x/\sqrt{x^2+1},\{\sqrt{x+1}\}/x(\log x{)}^2$$ according to the rapidity with which they tend to zero as $x\to \mathrm{\infty }$.

7. Arrange $$x\log \log (1/x),\sqrt{x}/\{\log (1/x)\},\sqrt{x\sin x\log (1/x)},(1–\cos x)\log (1/x)$$ according to the rapidity with which they tend to zero as $x\to +0$.

8. Show that $$D_x\log \log x=1/(x\log x),D_x\log \log \log x=1/(x\log x\log \log x),$$ and so on.

9. Show that $$D_x(\log x{)}^{\alpha }=\alpha /\{x(\log x{)}^{1-\alpha }\},D_x(\log \log x{)}^{\alpha }=\alpha /\{x\log x(\log \log x{)}^{1-\alpha }\},$$ and so on.