For instance how do we know that something weird doesn’t happen at some insanely large number?
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I had a calculus teacher explain an infinity graph like this and it has always helped.
Imagine you are standing in an endless hallway. You have a flashlight and shine it directly on the wall immediately next to you. This is the “lowest” point on the graph of “how far does the beam go before touching the wall”. Now start rotating to the left or right and the flashlight beam will still be hitting the wall, but at accelerating distances compared to your constant rotation. The point on the graph of “how far until the beam touches the wall” increases exponentially. There will come a point where your beam suddenly no longer touches the wall, but instead travels to infinity. Nothing special, just a mathematical representation showing that the beam will not touch. You continue to rotate and the beam comes back to you in the same pattern on the other wall.
It has been a lot of years since that class, but I THINK that describes an asymptote.
Now, if you’re asking what happens on the infinity side of where the light travels to, like what is way over there that we can’t know about, I don’t know. Science fiction explores that topic pretty thoroughly.
You say a sequence goes to infinity if for any number you pick, the sequence has a larger number.
So let’s use natural numbers, 1,2,3,4….
You pick a any number you want, say 999. I can then say at position 1000, in the sequence is 1000 which is greater than 999.
You would generalise it to be for any number e, there is a number e+1 in the sequence that is greater than it. So no matter what number you can think of, there is always a number in the sequence greater than it
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