I saw the following definition on youtube.
Let $f$ be a function defined on some interval $(a, +\infty)$
$\lim\limits_{x \to +\infty} f(x)= +\infty$ (1)
means $\forall N > 0$ $\exists M > 0$ such that $x > M$ implies $f(x) > N$.
Is this a correct and general definition for equation (1)? For example, the function $f$ defined below doesn't satisfy the above definition, but it is obviously divergent, as $x \to +\infty$.
$f(x) \equiv \log(x) \sin^2(x)$, $x \in (0, +\infty)$