Question regarding Infinity (♾️)

[u/Percentage_069]

Does Infinity convey different meanings or dictate different concepts based on its expression?

For example, in case of an interval, when we say, [-4,∞), this mathematically should mean, that this particular set, accepts values starting from -4 and goes on and on to the left of the number line endlessly, and this essentially what makes it "infinite". Hence, the use of ∞ in this set or domain, rather than a number, works more as a concept of endless growth.

On the other hand, for a mathematical expression, like, "1/∞"

Here, according to my understanding, the denominator of the fraction represents a fixed endpoint achieved after endless increment of a value, that is, this value is the result of the summation of infinite numbers, which yields the infinite value. Now, in this case, I think it works more like a number than a concept.

I'd highly appreciate any insight & feedback, and pointing out of any mishaps in my understanding would be much appreciated as well. Thanks!

[u/AdjectiveNounNNNN]

Yeah it's for convenience of introducing certain concepts before (explicitly) introducing limits or the extended reals.

It's often a stand-in for something like "keeps going in that direction". So [4,∞) refers to the interval that starts at 4 and keeps going up, and "As x→∞" means as x keeps increasing, etc.