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I'm taking a uni Calc. 1 course, and have come across something I have never seem before while reading on the arithmitic laws of infinite limits. One law states:

$$ 1/0^+=\infty $$

and another law states: $$ 1/\infty = 0^+ $$

What does $0^+$ mean?

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    $\begingroup$ It means that the denomiator function goes to zero and it is positive near the point where we compute the limit. $\endgroup$ Dec 12, 2021 at 19:46
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    $\begingroup$ “I have come across…” Where? In your textbook? Lecture notes? Somewhere on the internet? $\endgroup$ Dec 12, 2021 at 19:46
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    $\begingroup$ It looks like misleading notation to me, so without more context, it's very hard to guess. $\endgroup$
    – Lee Mosher
    Dec 12, 2021 at 19:46
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    $\begingroup$ Then give us the context from the book. As it is, it seems to be an abuse of notation. It could mean that if you are commuting $\lim_{x\to0^+} f(x),$ it is the same as $\lim_{y\to \infty} f(1/y).$ Basically, $=$ here is the abuse of notation, since these “laws” really say something about variable replacement in limits. $\endgroup$ Dec 12, 2021 at 19:54
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    $\begingroup$ What’s the name of the textbook? I’d like to see a picture of a page of the book which uses this notation. $\endgroup$
    – littleO
    Dec 12, 2021 at 19:59

2 Answers 2

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Assume that we want to compute $$L=\lim_{x\to a}\frac{1}{f(x)}$$ with $$\lim_{x\to a}f(x)=0.$$

So $$L=\frac 10$$ and we cannot say neither $L=-\infty $, nor $ L=\infty $

But, if the function $ f $ is positive near the point $ a$, then we write $$L=\frac{1}{0^+}=+\infty$$ For example, we have $$\lim_{x\to 1^-}\frac{1}{\ln(2-x)}=\frac{1}{0^+}=+\infty$$ and if $ f $ is negative then $$L=\frac{1}{0^-}=-\infty$$

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    $\begingroup$ It might be worth using the term “form” here. A limit of the “form” $\frac1{0^+}$ is $+\infty.$ We most often talk about the “indeterminate forms.” but these are examples of forms which are “determinate.” $\endgroup$ Dec 12, 2021 at 20:20
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If you divide $1$ by a very small positive number, you get a very large positive number.

If you divide $1$ by a very large positive number, you get a very small positive number.

That is, $0^{+}$ symbolically represents the concept of a small positive number.

You could consider $0^{-}$, representing a very small negative number. And then you'd have similar "laws" like $\frac{1}{0^{-}}=-\infty$.

Possibly, these "laws" are shorthand ways to remember something like "if $\lim_{x\to a} f(x)=0$ and $f(x)>0$ for all $x$ sufficiently close to $a$, then $\lim_{x\to a} \frac{1}{f(x)}=\infty$." That is a mouthful, and once you understand it, you can agree with the sentiment that $\frac{1}{0^{+}}=\infty$. Even though that "law" is using notation differently than most standards.

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