the notation of ``infinitly close to'' a point

Question

I seems to be taught before that in $\int_{a^+}^{b^-} f(x)dx$, the notation of $b^-$ mean a point infinitly close to x=b point from the left. Similarly for the $a^+$ (from the right). Am I right? if yes, what's the name of this notation rule? If not, what's the correct notations? Thanks!

Answer 1

Adding + above a number in limit means that we are approaching that value from positive side( right side) whereas - means approaching from left side.

Answer 2

No, there's no such thing as a point infinitely close to another point.

I've never seen the notation $\int_{a^+}^{b^-}$. If I did see that notation I can't imagine what it would mean other than $\int_a^b$. The notation $a^{\pm}$ comes up in limits, not integrals! $\lim_{x\to a^+}f(x)$ means the right-hand limit of $f$ at $a$.

The notation can be misleading. In particular, the $x\to a^+$ does not mean that $x$ is approaching some number $a^+$; you should parse it as $(x\to a)^+$, not $x\to(a^+)$.

(Ok, I thought of what the notation $\int_{a^+}^{b^-}$ might mean. Maybe it means $$\lim_{\alpha\to a^+}\lim_{\beta\to b^-}\int_\alpha^\beta.)$$