Negating definition of limit as x approaches infinity of a function

Question

Question

How do I negate the definition of limit as x approaches infinity of a function?

Answer 1

First case

$$\lim_{x\to -\infty}f (x)=L\in \Bbb R $$

the negation is $$(\exists \epsilon>0) \; \;( \forall A <0)\;\;(\exists x\in \Bbb R) \;\;:$$ $$ x <A \; \land\; |f (x)-L|\ge \epsilon $$

Second case

$$\lim_{x\to -\infty}f (x)=+\infty $$

the negation is $$(\exists B>0) \;\; (\forall A <0) \;\;(\exists x\in \Bbb R) \;\;:$$ $$x <A \;\;\land f (x)<B $$

The third for you.

Answer 2

Suppose we have a function $f: \mathbb R \to \mathbb R$ and $L\in \mathbb R$.

Then $L$ is said to be the limit of $f(x)$ as $x$ goes to infinity if and only if:

$\forall \epsilon \in \mathbb R: [\epsilon \gt 0 \implies \exists m\in \mathbb R: \forall x\in \mathbb R: [m\lt x \implies |f(x)-L|\lt \epsilon]].$

And it is false that $L$ is the limit of $f$ as $x$ goes to infinity if and only if:

$\exists \epsilon \in R: [\epsilon \gt 0 \land \forall m\in \mathbb R: \exists x\in \mathbb R: [m\lt x \land |f(x)-1| \ge \epsilon]].$