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From what I know, limits only exist if both side limits exist and are equal:
$${\lim_{x\to a}f(x) = L}$$ $$if$$ $${\lim_{x\to a^+}}f(x) = {\lim_{x\to a^-}f(x) = L}$$

But can this be applied to limits at infinity? In that case: $${\lim_{x\to \infty}f(x) = L}$$ $$if$$ $${\lim_{x\to +\infty} f(x)} = {\lim_{x\to -\infty}f(x) = L}$$

Is this correct or ${\lim_{x\to \infty}f(x)}$ should be taken as ${\lim_{x\to +\infty}} f(x)$ ?

Nick_17
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    Your definition for finite limits is wrong. When people write things like $\lim_{x\to a_+}$ they mean "the limit as $x$ approaches $a$ from the right. There is no condition at all on what happens near $-a$. – lulu Aug 31 '17 at 20:50
  • For $+a$ and $-a$, I guess you meant $a^+$ and $a^-$, which have very different meaning: https://en.wikipedia.org/wiki/One-sided_limit –  Aug 31 '17 at 20:52
  • Yes, I meant that :s – Nick_17 Aug 31 '17 at 20:53
  • What book are you reading? –  Aug 31 '17 at 20:53
  • Calculus by Tom M. Apostol – Nick_17 Aug 31 '17 at 20:55
  • Now that you have fixed the typo, can you answer your own question now? Note also that $\infty$ is not a real number. The notation $\lim_{x\to\infty}f(x)$ has a very specific meaning, either in terms of topology on ${\bf R}\cup{\infty}$ or the "$\epsilon$-M" language. What is it in Apostol's book? –  Aug 31 '17 at 20:58
  • You mean that because $\infty$ is not a real number that rule cannot be applied? In that case both limits are independent and ${\lim_{x\to \infty}f(x)}$ should be evaluated only with positive $x$? – Nick_17 Aug 31 '17 at 21:03
  • Infinity can only be approached from one side. – Qiaochu Yuan Aug 31 '17 at 21:07
  • In Apostol's book it is clearly stated that limits at positive infinity are denoted like this ${\lim_{x\to +\infty} f(x)}$ So that confuses me when I look problems with the following notation ${\lim_{x\to \infty}f(x)}$ – Nick_17 Aug 31 '17 at 21:12
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    You are confusing notation. "positive infinity" is written $\infty$ or $+\infty$ and "negative infinty" is written $-\infty$. The notation $a^+$ and $a^-$ means approaching via values larger than $a$ or via values less than $a$. The notation $\infty^+$ or $-\infty^-$ would be meaningless as we can't approach infinity from values more than infty (or less than neg. infty). And $\infty^-$ and $-\infty^+$ would be unnesc as we can only approach infty from those directions. – fleablood Aug 31 '17 at 23:16
  • $+\infty$ and $\infty$ are considered the same. Notice that $\lim_{x\to 5^-}$ means the "left-hand" limit. That is as values of x that are less than postive 5 approach 5. But $\lim_{x\to -5}$ means something entirely different. That means the values of x as they approach negative 5. – fleablood Aug 31 '17 at 23:21

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