using logical quantifiers to write that f approaching infinity DOES NOT tend to infinity

Question

Is this the same as writing that the limit of f as f approaches $\infty$ is L? i.e.: $\forall \space \epsilon > 0 \space \exists \space c \space \forall \space x>c : |f(x) - L|< \epsilon$

Answer 1

Saying that $f(x)\not\to\infty$ as $x\to\infty$ is not the same thing as saying $f(x)\to L$ as $x\to\infty$. $f$ may not tend to any limit. The first statement with quantifiers

$$\exists M>0\,\forall r\in\Bbb R\,\exists x>r\,f(x)\le M.$$

To see why this is, consider the statement $\lim_{x\to\infty}f(x)=\infty$. Formally this is

$$\forall M>0\,\exists r\in\Bbb R\,\forall x>r\,f(x)>M.$$

This can be informally translated as no matter how large an $M$ we pick, we can always find a real number $r$ so that the value of $f$ exceeds $M$ for any real number larger than $r$.

When the sentence is negated each $\forall$ changes to $\exists$ and vice versa and the statement after the quantifiers is negated. That's how I got the first statement.

Informally it says that there is an $M$ so that no matter how large we pick $r$ there will be an $x>r$ for which the value of $f$ is no bigger than $M$.