# Arrow in limit operator

In this Wikipedia article, I see a limit operator such as in:

$$\lim_{x \searrow 0} \frac{e^{-1/x}}{x^m}=0\,\,;\,\,\,\, m\in \mathbb{N}$$

I am assuming that the downward pointing arrow indicate the limit as $x$ approaches $0$ from the positive direction? Is this conventional? I've seen both $\displaystyle\lim_{x \rightarrow 0⁺}$ and $\displaystyle\lim_{x \downarrow 0}$, but never before $\displaystyle\lim_{x \searrow 0}$

• Yes, all of those notations are the same. This is more often seen in analysis rather than calculus, especially in the context of sequences which are decreasing. Commented May 2, 2011 at 1:00
• To be clear, x→0+ means approaches from the right, but using any sequence of positive numbers converging to 0 you'd like. Whereas x↘0 means approaches using any decreasing sequence of (positive) numbers converging to 0. I doubt there is much difference. Commented May 2, 2011 at 1:36

## 1 Answer

Yes, it means that considers decreasing sequences that converge to 0.

I've only once worked with someone who preferred to use the $\searrow$ and $\nearrow$ notation, but it's a good notation in the sense that it takes only a moment to become completely confident in what it means. That's one of the best parts of writing out math, I think - we can invent our own notation so long as it follows intuitive guidelines (rather than very strict, traditional guidelines) in many cases.

• What about an arrow straight down? en.wikipedia.org/wiki/C0-semigroup#Infinitesimal_generator I feel the opposite, trying to learn is very frustrating when everyone invents their own notation and leaves little quirks unexplained, especially when they're trying to communicate/teach. When you're learning new content, you won't know when you can trust your intuition, you might be learning the notation for the first time, and you won't know what's important to the syntax or just your professor feeling artistic. Commented Dec 15, 2020 at 13:15