How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$?

Question

How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$

I suppose it's $\forall K\in\Bbb{R},\ \exists N\in\Bbb{R},\ x\gt N\implies f(x)\gt K$.

Could someone correct it?

Answer 1

Yes that is the correct definition of $ \lim_{x \to \infty} f(x) = \infty $

Answer 2

Your definition is technically wrong, because you are missing the universal quantification of $x$. Without the quantifier, $x$ would be a free variable, which conventionally is interpreted essentially the same as an external quantification, which is on the outside. This is of course different from having the quantifier on the inside, because you cannot switch $\forall$ and $\exists$. The correct one is: $\def\rr{\mathbb{R}}$ $\def\imp{\rightarrow}$

$\forall k \in \rr\ ( \exists n \in \rr\ ( \forall x \in \rr\ ( x > n \imp f(x) > k ) ) )$