What does it mean to say $f(x) \sim g(x)$, i.e. $f(x)$ behaves like $g(x)$ when $x \to \infty$?

Question

If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$, then $f$ grows faster than $g$. Same if $\lim_{x\to\infty} \frac{g(x)}{f(x)} = 0$.

Would $f$ behave like $g$ if $\lim_{x\to\infty}\frac{f(x)}{g(x)} = 1$ ?

Answer 1

Yes, by definition,

$$f(x) \sim g(x) \quad \mbox{as} \quad x \to \infty$$

means

$$\lim_{ x \to \infty} \frac{f(x)}{g(x)} = 1$$