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.

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    $\begingroup$ Your definition is correct. you just replaced $A $ by $\delta $. $\endgroup$ – hamam_Abdallah Sep 2 '17 at 14:37

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 $\delta$.

  • $\begingroup$ +1. But it's still worth pointing out that the choice of $\delta$ to represent small quantities and $L$ to represent large numbers is a matter of tradition and habit (and our psychology, maybe?). Yes, we all stick to these conventions as it's much more convenient. But as long as all signs and symbols are right, the definition is correct regardless of the letters used. $\endgroup$ – zipirovich Sep 2 '17 at 15:11
  • $\begingroup$ @zipirovich agreed. But it's about communicating the idea. So we try not to confuse readers of the definition. $\endgroup$ – Henno Brandsma Sep 2 '17 at 15:13
  • $\begingroup$ Just to make sure: I wasn't arguing with you! :-) But it seems to me that the OP has doubts whether the definition as he/she stated is correct. So I wanted to state clearly that yes, it is correct. It's just that we prefer using certain letters in certain contexts. It doesn't change anything mathematically, but helps us understand each other better. $\endgroup$ – zipirovich Sep 2 '17 at 15:16

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