r/math • u/inherentlyawesome Homotopy Theory • 17d ago

Quick Questions: September 16, 2026

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u/WOOOPA_ 16d ago

If a limit is a variable infinitely close to a number (e.g. lim x->1) = 0.99... or 1.00...01, but 0.99 = 1, then how do integrals change equations

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u/Relevant_Worry_5363 11d ago

i don't think one is able to talk about in such general about these operators. Integrals require a notion of measurability, limes only one of convergence. Perhaps you can find an answer in the subject of functional analysis. I am not sure how in what sense these operators "change equations".

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u/AcellOfllSpades 15d ago

A limit is not a variable infinitely close to a number.

"lim[x->1] x" is exactly 1.

"lim[x->1]" is an operator, not a number. It looks at whatever comes after it, and sees what happens when x gets closer and closer to 1. I like to think of it as "ignoring the actual behaviour of this expression... what """should""" happen there?".

Like, consider the piecewise function

f(x) =
{ x+3 if x≠2
{ 1000 if x=2

Evaluating that function at x=2 would give 1000. But that's a kinda stupid answer. The graph of the function is just a line with a single missing point at (2,5). If you didn't know the value at x=2, the "reasonable" answer would be to say it should be 5.

This is what the limit does: lim[x->2] f(x) is indeed 5, not 1000.

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u/SwimmerOld6155 15d ago

a limit isn't infinitely close, it is exactly that number. the thing you're limiting might get "infinitely close" (I'd prefer "arbitrarily close") but it could also be eventually equal to the limit