this post was submitted on 16 May 2024
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Almost. 1/x approaches infinity from the positive direction, but it approaches negative infinity from the negative direction. Since they approach different values, you can't even say the limit of 1/x is infinity. It's just undefined.
it is possible to rigorously say that 1/0 = β. this is commonly occurs in complex analysis when you look at things as being defined on the Riemann sphere instead of the complex plane. thinking of things as taking place on a sphere also helps to avoid the "positive"/"negative" problem: as |x| shrinks, 1 / |x| increases, so you eventually reach the top of the sphere, which is the point at infinity.
https://en.wikipedia.org/wiki/Division_by_zero#Floating-point_arithmetic
10/0 β lim x->0+ 10/x
Or in other words, the thing you keep quoting does not apply in this case. Any number divided by zero is undefined, not positive infinity (or negative infinity for that matter).
To be fair, it turns out not all environments implement floating-point arithmetic by the IEEE spec, meaning division by 0 can produce different results depending on where you run it. So in C++ float division by zero is undefined: https://stackoverflow.com/questions/42926763/the-behaviour-of-floating-point-division-by-zero
But I'm fairly sure (note: based on literally no research) that most environments today will behave like the IEEE spec.
It's undefined in math, but not floating point arithmetic