Which theorems for finite limits remain true for infinite limits? The algebra of limits follows. What are the other properties and theorems that follow? How to check? On which theorems should I check?


closed as unclear what you're asking by Lord Shark the Unknown, Brevan Ellefsen, Alex Ortiz, Morgan Rodgers, Mark Bennet Aug 23 '17 at 20:43

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  • $\begingroup$ A lot of things. You just have to take care in defining what a limit at infinity means. A common real analysis definition is that, given $\epsilon > 0$, $\exists N\in\mathbb{N}: |a_n-a|<\epsilon\, $ for all $n \ge N$. There are nicer ways to rigorously define this, such as One-Point Compactification for both real and complex numbers, but since you clearly have little introduction to this topic that's probably a bit beyond what you should worry about for now. $\endgroup$ – Brevan Ellefsen Aug 23 '17 at 16:30
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    $\begingroup$ Unclear. What means "infinite limits"? Limits with variable $\to\infty$ or limits with function $\to\infty$? $\endgroup$ – Martín-Blas Pérez Pinilla Aug 23 '17 at 17:00
  • $\begingroup$ Obviously. With functions $\endgroup$ – user421114 Aug 23 '17 at 17:10
  • $\begingroup$ @Mathhacker is this a good example. $\endgroup$ – hamam_Abdallah Aug 23 '17 at 21:43

If, near $x=a $,

$$f (x)<g (x) $$ then $$\lim_af (x)\le \lim_ag (x) $$

  • $\begingroup$ Really? What if $f \to 0$ and $g \to \infty$? $\endgroup$ – Hans Lundmark Aug 23 '17 at 19:21
  • $\begingroup$ @HansLundmark Thnx. I edited new. $\endgroup$ – hamam_Abdallah Aug 23 '17 at 21:42