Ambiguity of saying that a function diverges to infinity

[Bohandas]

It occurs to me that saying that a function goes to infinity seems somewhat ambiguous due to the fact that there are different sizes of infinity.

In particular the value of 2^x remains strictly higher than x^2 even when x is infinite. If I understand correctly, 2^x will always be one beth number higher.

Is describing them both simply as diverging to infinity an artifact of undergraduate math being simplified and the distinction actually is made in more advanced examinations of sequences and functions? or is it because infinity is imagined in these scenarios as being approached but never actually reached? Or it it conversely a case of imagining that absolute infinity has been reached?

[NichG]

Yeah, different asymptotics can matter. You see practical examples of this in things like contour integrals of complex functions, and in general the whole stuff with residues. Basically, multiply a function by (z-z0), does it still diverge at z0? How about if you multiply by (z-z0)^2? Etc.

Also shows up in ambiguous limits. You could have one function going to infinity, another going to zero, but their product could be zero, finite, infinite, etc. lim x->0 sin(x)/x for example is finite and nonzero, but lim x->0 1/x of course goes to infinity. If on the other hand you had lim x->0 sin(x)/x^2, that diverges.

[ahyangyi]

Yep, not all functions that diverges to infinity are equal. They have different sizes. When that matters, asymptotic analysis gives you the necessary terms and notations.

f(x) = 2^x is asymptotically larger than g(x)=x^2 as x -> infinity.
g(x) = x^2 is asymptotically equivalent to h(x)=(x+10000)^2 as x -> infinity.

And when the function itself doesn't even matter, only the asymptotic "level" matters, we often use Big O notation.

... and, well, since this is playground, you could apply all these stuff to the "linear fighter, quadratic wizard" trope...

Hope that helps.

[akma]

Generally when people use x it is implied x is a real number, as in x member of R. Types of infinities are not really relevant in those cases.

[Willie the Duck]

Generally when people use x it is implied x is a real number, as in x member of R. Types of infinities are not really relevant in those cases.

And infinities tend to have the same upward bound, so the diverging to infinity part isn't usually particularly ambiguous.

[Radar]

And infinities tend to have the same upward bound, so the diverging to infinity part isn't usually particularly ambiguous.

This - the statement that a given function diverges to infinity is not ambiguous as for example both 2^x and x^2 go in the same direction and do not stop at some finite number. How fast they grow is a completely different question. This would be probably more visible if we consider functions divergent at some finite argument value like the mentioned 1/x and for example 1/x^2. For any argument value ratio of divergent functions can behave very differently, but if you plot those functions it will be clear that both go toward the same direction close to 0. After all infinity^2 = 2^infinity = infinity (note: this is different from set cardinality calculations).

[halfeye]

In particular the value of 2^x remains strictly higher than x^2 even when x is infinite. If I understand correctly, 2^x will always be one beth number higher.

Strictly pedantic and minor quibble (particularly when the main concern is behavour near infinity), between 0 and 2 these two functions do not behave as described:

2^0 =1, 0^2 = 0, which is as said,

2^1 =2, 1^2 = 1, again as said,

2^2 = 2^2 = 4 not as said,

2^1.5 = ?, 1.5^2 = 2.25, which I'm very suspicious is the reverse of what was said.

[DavidSh]

Strictly pedantic and minor quibble (particularly when the main concern is behavour near infinity), between 0 and 2 these two functions do not behave as described:

2^0 =1, 0^2 = 0, which is as said,

2^1 =2, 1^2 = 1, again as said,

2^2 = 2^2 = 4 not as said,

2^1.5 = ?, 1.5^2 = 2.25, which I'm very suspicious is the reverse of what was said.

To me, it looks like the cross-over region is from x=2 to x=4.
As you wrote, 2^2 = 4 = 2^2.
Also 2^4 = 16 =4^2.
But 2^3 = 8 < 9 = 3^2.

[Radar]

To me, it looks like the cross-over region is from x=2 to x=4.
As you wrote, 2^2 = 4 = 2^2.
Also 2^4 = 16 =4^2.
But 2^3 = 8 < 9 = 3^2.

You are correct: 2^x < x^2 for 2<x<4

[Asmotherion]

My question is, does it matter in the physical world?

I mean we have no way of measuring infinity, so in a way it's still a supernatural phenomenon to us.

Is there a point where the universe ends? Is the Universe some sort of finite bubble? We shall probably never discover an answear to that, even if we develop our technology to light speed rockets or even warp technology, if the answear is infinite.

So, even if an infinity is, on paper, larger than an other infinity, let's say ∞^2, ∞^∞
the actual result will be the exact same, an infinity.

it's the same as multiplying with 0. On paper, it seems like 10*0 would be larger than 5*0 but the result is the exact same aka 0.

At least that's my viewpoint.

[Fat Rooster]

It occurs to me that saying that a function goes to infinity seems somewhat ambiguous due to the fact that there are different sizes of infinity.

In particular the value of 2^x remains strictly higher than x^2 even when x is infinite. If I understand correctly, 2^x will always be one beth number higher.

Is describing them both simply as diverging to infinity an artifact of undergraduate math being simplified and the distinction actually is made in more advanced examinations of sequences and functions? or is it because infinity is imagined in these scenarios as being approached but never actually reached? Or it it conversely a case of imagining that absolute infinity has been reached?

It is no more or less ambiguous than a word like 'fruit'. It means something precise and important, but we can get far more specific, and often have to. Whether a distinction is made when comparing two sequences is entirely dependent on why and how you are comparing the two sequences. Big O notation is the best known tool set past just describing it as diverging, but even that still leaves information that may be relevant for certain purposes, so is in a sense ambiguous.

[Willie the Duck]

[FONT=arial]My question is, does it matter in the physical world?

I mean, on some level this is a bunch of underchallenged nerds spinning their wheels --possibly with a side order of making sure everyone else knows they've read Cantor (like everyone else who would frequent such a forum). There doesn't need to be a point.

Regarding multiple infinities or functions diverging to infinity in the physical world, generally no. Math is a human construct which sometimes overlaps with the physical world. Sometimes these 'esoteric musing'-style maths do lead to some kind of breakthrough in applied knowledge like cryptography, computer science, physics, or computer science. More often than not, though, they are more just byproducts of intellectual curiosity.

Functions, though, are important to (the modelling of, and thus exploration of) all sorts of real world phenomena. That a function does or doesn't diverge to infinity often is an important factor in analyzing the function's applicability, even though we'll never have the infinity iteration in an applied setting.

The size of the universe being infinite or not, probably in and of itself will not inform anything in our lives. However, the answer to the question probably tells us a lot about how the universe formed (and out of what). That has application to, if not our immediate needs, at least to our understanding of stellar mechanics and why our local (visible) stellar neighborhood looks like it does.

[NichG]

My question is, does it matter in the physical world?

I mean we have no way of measuring infinity, so in a way it's still a supernatural phenomenon to us.

Is there a point where the universe ends? Is the Universe some sort of finite bubble? We shall probably never discover an answear to that, even if we develop our technology to light speed rockets or even warp technology, if the answear is infinite.

So, even if an infinity is, on paper, larger than an other infinity, let's say ∞^2, ∞^∞
the actual result will be the exact same, an infinity.

it's the same as multiplying with 0. On paper, it seems like 10*0 would be larger than 5*0 but the result is the exact same aka 0.

At least that's my viewpoint.

It absolutely matters, because the asymptotic behavior of functions also applies when arguments get very large or small, not just infinite or actually zero. And those asymptotic behaviors are often more mathematically or theoretically tractable than the full behavior of the function or physical system.

So for example our ability to understand fluids and aerodynamics - it's easier to start at infinite Reynolds number and then patch in finite Reynolds number effects than to work directly at the real values in a lot of cases. The way you deal with that patching involves methods like matched asymptotics, basically fitting two local approximations to each other such that their behaviors at infinity match. That matching in turn determines the drag force.

Or with quantum mechanics, perturbation theory is an order-by-order approximation that can let you do things like calculate the induced separation of energy levels that things like MRI rely on. When you have degenerate energy levels due to symmetries of the system, you need to 'cancel ' out an infinity correctly to proceed. More broadly, renormalization is the process of systematically matching infinitely diverging terms order by order in order to stabilize diverging approximations - that's the method we use for calculating things like the behavior of magnets or fluids near their critical points.

[MetroAlien]

There is no contradiction between these two statements:
A: neither function has a "maximum value", i.e. it's infinite
B: when evaluated at any given point, one function is consistently larger than the other

notice the bold part:
to evaluate the function you need to localise it.
From that moment on, any talk about "infinity" becomes irrelevant.
So what if the function will never max out?
In this instance we're only interested in this specific point, which is by definition finite.

[halfeye]

There is no contradiction between these two statements:
A: neither function has a "maximum value", i.e. it's infinite
B: when evaluated at any given point, one function is consistently larger than the other

notice the bold part:
to evaluate the function you need to localise it.
From that moment on, any talk about "infinity" becomes irrelevant.
So what if the function will never max out?
In this instance we're only interested in this specific point, which is by definition finite.

Except in the case given, that bolded part was in one small region false, so "consistently" or "always" are in this case also false.

There can be examples where statements including "consistently" or "always" are true, but the case given accidentally wasn't one of those.