Textbook Typo regarding Growth of Functions

Question

According to Rogawski's Calculus - Early Transcendentals (3rd Ed.), it states

To indicate that $f(x)$ grows faster than $g(x)$, we use the notation $f(x)\ll g(x)$. For example, $x^2 \ll x$ because $$\lim_{x\rightarrow\infty}\frac{x^2}{x}=\lim_{x\rightarrow\infty}x=\infty$$

Then it gives an example,

Which of $f(x)=x^2$ and $g(x)=x\ln{x}$ grows faster as $x\rightarrow\infty$?

Solving the problem,

$$\lim_{x\rightarrow\infty}\frac{x^2}{x\ln{x}}=\lim_{x\rightarrow\infty}\frac{x}{\ln{x}}=\lim_{x\rightarrow\infty}\frac{1}{x^{-1}}=\lim_{x\rightarrow\infty}x=\infty$$

And he concludes that $x\ln{x}\ll x^2$

So which is is? I would assume the faster growing function would be to the right of the relation, and so if $f$ grows faster than $g$, it should be that

$$g\ll f$$

Answer 1

You're right. The faster growing function should be in the right of the relation. In this case, $g ≪ f$ indicates $f$ is a faster-growing function.

Quoting Kassabov and Pak:

Let $f, g :\mathbb{N} → \mathbb{N}$ be two integer functions, such that $f(n), g(n) → ∞$ as $n→∞$. We write $f ≪ g$ if $f(n) < g(cn)$, for some $c > 0$ and infinitely many $n ∈ \mathbb{N}$.