Example 4

For $x$ real tending to $+\infty$ the family $\mathscr{E}$ of functions of the form $\exp(p(x))$, where $p$ runs through the set of polynomials without constant term (with real coefficients), is a comparison scale: it suffices to remark that the quotient of two functions in $\mathscr{E}$ is again in $\mathscr{E}$, and that a function $\exp(p(x))$ must tend either to 0 or $+\infty$ if $p \neq 0$; indeed, $p(x) \sim \alpha x^n$ where $n > 0$ and $\alpha \neq 0$; if $\alpha > 0$ then $p(x) > \frac{1}{2} \alpha x^n$ for $x$ sufficiently large; if $\alpha < 0$ then $p(x) < \frac{1}{2} \alpha x^n$ for $x$ sufficiently large; in the first case $\exp(p(x))$ tends to $+\infty$, in the second to 0.