# What does $0^+$ mean [closed]

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?

• It means that the denomiator function goes to zero and it is positive near the point where we compute the limit. Dec 12, 2021 at 19:46
• “I have come across…” Where? In your textbook? Lecture notes? Somewhere on the internet? Dec 12, 2021 at 19:46
• It looks like misleading notation to me, so without more context, it's very hard to guess. Dec 12, 2021 at 19:46
• 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. Dec 12, 2021 at 19:54
• What’s the name of the textbook? I’d like to see a picture of a page of the book which uses this notation. Dec 12, 2021 at 19:59

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 $$L=\frac{1}{0^-}=-\infty$$

• 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.” Dec 12, 2021 at 20:20

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.