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

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    $\begingroup$ 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 $-a$. $\endgroup$ – lulu Aug 31 '17 at 20:50
  • $\begingroup$ For $+a$ and $-a$, I guess you meant $a^+$ and $a^-$, which have very different meaning: en.wikipedia.org/wiki/One-sided_limit $\endgroup$ – Jack Aug 31 '17 at 20:52
  • $\begingroup$ Yes, I meant that :s $\endgroup$ – Nick_17 Aug 31 '17 at 20:53
  • $\begingroup$ What book are you reading? $\endgroup$ – Jack Aug 31 '17 at 20:53
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    $\begingroup$ 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. $\endgroup$ – fleablood Aug 31 '17 at 23:16

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