Exact definition of limits going to infinity

Question

I have been reading that the $\lim_{x\rightarrow \infty}f(x)=a$ looks like $\forall \epsilon>0\;\exists\delta\in\mathbb R: \;x>\delta\implies |f(x) - a|\leq\epsilon$ from a previous asked question on this site. However, another user commented on this post that limits going to infinity do not use the $\epsilon-\delta$ definition as the limit goes to infinity. Is this correct? If so, I am curious to see what the definition/axiom would be for a limit going to infinity like above.

Answer 1

It's not common to use $\delta>0$ for limits to infinity. The $\delta$ suggests a small real number. For a limit as $x$ tends to $p$ we need to be able to approximate $f(p)$ better and better by going closer and closer to $p$ and points close to $p$ are in $(p-\delta, p+\delta)$ for smaller and smaller $\delta$.

But a neighbourhood of "infinity" is different: we consider bigger numbers to be closer to infinity. So corresponding to $(x-\delta, x+\delta)$ neighbourhoods of $p$ we use neighbourhoods of infinity of the form $(L,\infty) = \{x \in \mathbb{R}: x > L \}$.

The approximation property then becomes

$$ lim_{x \to \infty} f(x) = a \leftrightarrow \forall \varepsilon>0: \exists L > 0: (\forall x > L): |f(x) - a| < \varepsilon$$

So the same as you stated but with $L$ instead of $\delta$.