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I'm not an English native speaker and I don't know how to pronounce $0^+$ and $0^-$.

Could you help me, please?

I'm referring to the symbols that appear for example in the limits $lim_{x\to 0^+}f(x)$ and $lim_{x\to 0^-}f(x)$. Thank you in advance for the help.

John N.
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1 Answers1

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Those are right-sided and left-sided limits. You can say, as $x$ approaches $0$ from the right side, $f(x)$ approaches $L$. You can also say the limit of $f(x)$ as $x$ approaches $0$ from the right side.

Of course, the natural association of being to the left and right of a value depends on the dialect and language. You could use the notion of being below or above a value in place of mentioning right or left.

Andrew Li
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  • Thank you for the answer, but that's not what I asked. I wanted to know how to read out the two symbols $0^+$ and $0^-$ alone, not the limits. The limits were just examples that I wrote to be sure that all the people reading this post understand my question. – John N. Feb 28 '18 at 04:10
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    @JohnN. But where do you use them besides limits? It all depends on the context, because the context gives it meaning. Without the context of a limit, the + and - are meaningless, because they are meant to represent the direction of approach, right? – Andrew Li Feb 28 '18 at 04:11
  • I'll give you an example. I want to calculate a limit and I get an indeterminate form, let's say $0^+/0^-$. In this case I don't know how to read these symbols and describe this indeterminate form. – John N. Feb 28 '18 at 04:26
  • @JohnN. Can you share the limit that gives you that? Can you explain what $0^+$ and $0^-$ even mean here? They don't convey anything to me. – Andrew Li Feb 28 '18 at 04:27
  • for example you could consider the limit lim_{x->1^+} (1-x)/(e^{x-1}-1) This should be an indeterminate form 0/0. But if one wants to be pernickety, the numerator would be 0^- (by this I mean that it approached 0 from the left), while the denominator is 0^+ (by this I mean that it approached 0 from the right). – John N. Feb 28 '18 at 04:41
  • @JohnN. But does that mean anything? It's still $0 \over 0$ (which you take L'Hopital). If you're taking a one-sided limit, then you're only observing the behavior of a function from one side, but you don't say as $x$ approaches $1$ from the right side, $f(x)$ approaches 0 on the right. You just say that $f(x)$ approaches 0 because the limit is already one-sided. – Andrew Li Feb 28 '18 at 04:52
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    @JohnN. Anyways, in your example it makes no sense to distinguish between $0^+$ and $0$ because limits gives a single value or infinity or don't exist. They don't give something such as "0 from the right" implied by $0^+$. You can call it zero-plus or "zero from the right" as in the comments but in your presented example it makes no sense to distinguish it and just $0$. – Andrew Li Feb 28 '18 at 05:28
  • I agree that isolating the symbols $0^+$ and $0^-$ from their proper context is deceptive, unless one uses the language very carefully. Although I like the comment of @MichaelMcGovern to say for example "zero from the right", although it only applies in the larger context of the symbols "$x \to 0^+$", and then the whole thing becomes "as $x$ approaches zero from the right". – Lee Mosher Sep 24 '18 at 15:28