What does the superscript plus/minus in limit notation really mean? I'm a beginner calc student, and I see limits in this context:
$$\lim_{x\to n^+}f(x)$$
In my class and in most videos I find online, it means "approaching $n$ from the right", but some people say it means "$x$ is a number infinitesimally larger than $n$", and vice versa for $n^-$.
Are these interchangeable, or is one correct?
 A: In the real numbers, there is no such thing as a value that is "infinitesimally larger" (or smaller) than another. There are constructions and systems like non-standard analysis that add infinitesimal values to the real numbers, but typically that's not what is going on unless explicitly stated.
The notation $\lim_{x \rightarrow a^+} f(x)$ means "the limit of $f(x)$ as $x$ approaches $a$ from above", or in other words it is the limit where you consider only values of $x > a$. On the real number line you can swap "from above" with "from the right", but that may get confusing if you're working in the Cartesian plane and taking a limit on $y$, for example.
Similarly, $\lim_{x \rightarrow a^-} f(x)$ would be considered "the limit of $f(x)$ as $x$ approaches $a$ from below", and then $\lim_{x \rightarrow a} f(x)$ exists if and only if the limits from above and below both exist, and are equal to one another.
A: infinitesimal refers to an extremely small value approaching zero. "infinitesimally larger" isn't a valid concept. The notation in which you listed is read as "the limit of f(x) as x approaches n from the right. You would be correct that if it was an - instead + that it would be from the left.
