## Infinity

For the discussion of math. Duh.

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idontknow4231
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### Infinity

Infinity has always bothered me, but also fascinated me. From holes in graphs in higher algebra, now to limits as x approaches infinity in calculus.

My question comes with ln(x). Everyone agrees that there is no limit as ln(x) approaches infinity, that it reaches infinity. But I was thinking about it the other day, and largest value my calculator can calculate, 9x10^99, roughly, was only equal to 230 when plugged into ln(x)!

This got me thinking, ln(x) grows so slowly, how can it ever really reach infinity? It seems like the distance between each whole number, represented by e^x, would get to be infinity long before ln(x) does. I know in the end infinity will win out, and I'll have to accept it, but has anyone else pondered this? Or can anyone just show me some way really quickly?

Just for fun, I checked on wolframalpha, and the ln(googol^googol) was only 2.5*10^102 or so, using over a googolplex.

I just can't comprehend it I guess.

t0rajir0u
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### Re: Infinity

All the logarithm does is basically bring down the exponent. So if you want to convince yourself that the logarithm really does go to infinity, use the values [imath]x = e, e^e, e^{e^e}, e^{e^{e^e}}, ...[/imath].

In general, just because something goes to infinity doesn't mean it does so quickly. For example, the inverse Ackermann function goes to infinity but is, for all practical purposes, at most 5.

lightvector
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### Re: Infinity

What it means for f(x) to approach infinity as x approaches infinity, is that for any bound M, no matter how large, f(x) will eventually exceed M (and remain larger than M thereafter).

So why does ln(x) approach infinity as x approaches infinity? Simple. For any M, ln(x) will exceed M once x exceeds e^M, since ln(x) is the inverse of e^x.

For instance, it becomes bigger than 3 starting after x = e^3, which is about 20.09. It becomes bigger than 1000 when x > e^1000. It will become bigger than 100^(100^(100^100)) once x > e^(100^(100^(100^100))). Clearly, it will surpass any finite bound.

Qaanol
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### Re: Infinity

If [imath]y = \ln(x)[/imath] then [imath]x = e^y[/imath].
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Syrin
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### Re: Infinity

That's just the normal infinity. That one's no fun. The fun ones are the cardinalities.

jestingrabbit
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### Re: Infinity

idontknow4231 wrote:stuff

By the change of base formula, ln(x) = log10(x)/log10(e). If 10k-1< x < 10k then k-1 < log10(x) < k, and so

log10(e)(k-1) < ln(x) < log10(e) k.

Now, C=log10(e) is just a number (about 2.30258509...) and if you look at that inequality, then you can pretty quickly work out that if x has k digits in its representation before the decimal point, then ln(x) will be a little less than C*k, so, you wrote that you could get a number with ln(x) = 230 in your calculator, and from this I know that you can get about 10^100 in your machine.

So that's really what's going on. ln is growing at about 2.3 times the number of digits to the left of the decimal point.

btw, from the fact that ln(ab) = b ln(a), you can pretty quickly come to a precise expression for ln(googol^googol), which is C*10^102. Not sure what happened to alpha on that one.
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Token
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### Re: Infinity

jestingrabbit wrote:btw, from the fact that ln(ab) = b ln(a), you can pretty quickly come to a precise expression for ln(googol^googol), which is C*10^102. Not sure what happened to alpha on that one.

I'm guessing the OP just misremembered the number after the decimal point. Wolfram Alpha gives the correct answer when I try it.
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andrewxc
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### Re: Infinity

OP: This is similar to the way that x = y^2 will yield an x-limit of infinity as y approaches infinity. The parabola will approach infinity in the x-direction "much more quickly" and with evermore increasing quickness than it will approach infinity in the y direction, but as the other posters stated, if you pick any number as your input, you will get a real, distinct output.

Also as was stated above, if ln(x) has a limit, then so should its inverse function, e^x.
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DavCrav
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### Re: Infinity

andrewxc wrote:Also as was stated above, if ln(x) has a limit, then so should its inverse function, e^x.

I might be wrong, but if [imath]\ln(x)[/imath] has a limit, wouldn't that mean that [imath]\mathrm{e}^x[/imath] attained infinity at a finite value, so wouldn't be defined after a certain point? A bit like [imath]\tan(x)[/imath].

hawkmp4
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### Re: Infinity

DavCrav wrote:
andrewxc wrote:Also as was stated above, if ln(x) has a limit, then so should its inverse function, e^x.

I might be wrong, but if [imath]\ln(x)[/imath] has a limit, wouldn't that mean that [imath]\mathrm{e}^x[/imath] attained infinity at a finite value, so wouldn't be defined after a certain point? A bit like [imath]\tan(x)[/imath].

Yes. If [imath]\lim_{x \to \infty} \ln(x) = c[/imath], where c is some finite number...
Put both sides to the power of e. Since e^x is continuous, you can move the limit out of the exponent on the right side and you have...
[imath]\lim_{x \to \infty} x = \mathrm{e}^c[/imath]
Which is clearly not true.
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minno
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### Re: Infinity

The annoying thing about infinity is that it really isn't a definite number. For example, as x -> infinity, both y = x and y = x^2 approach infinity, but x never approaches x^2. You can't treat infinity like it's the same number in all contexts.
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andrewxc
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### Re: Infinity

minno wrote:The annoying thing about infinity is that it really isn't a definite number. For example, as x -> infinity, both y = x and y = x^2 approach infinity, but x never approaches x^2. You can't treat infinity like it's the same number in all contexts.

Technically, x^2 approaches infinity faster than x, but there is no way to distinguish between them "at infinity", because they have no value.
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hawkmp4
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### Re: Infinity

What about [imath]\displaystyle\lim_{x \to \infty} \frac{x^2}{x}[/imath]? That gives a perfectly good distinction between the functions' behaviour as they grow without bound.
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t0rajir0u
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### Re: Infinity

Paradoxes only happen if you think of infinity as a number. It's much more satisfying, and much more relevant to modern mathematics, to think of infinity as a place. Different functions get there at different speeds. Why is that concept difficult to swallow?

Dason
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### Re: Infinity

t0rajir0u wrote:Paradoxes only happen if you think of infinity as a number. It's much more satisfying, and much more relevant to modern mathematics, to think of infinity as a place. Different functions get there at different speeds. Why is that concept difficult to swallow?

Hey guys. Party... be there. Where? Why it's at infinity of course. See ya there!
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AllSaintsDay
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### Re: Infinity

Dason wrote:
t0rajir0u wrote:Paradoxes only happen if you think of infinity as a number. It's much more satisfying, and much more relevant to modern mathematics, to think of infinity as a place. Different functions get there at different speeds. Why is that concept difficult to swallow?

Hey guys. Party... be there. Where? Why it's at infinity of course. See ya there!

You just know that 1/n (log n)( log log n) is going to show up "fashionably late."

mike-l
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### Re: Infinity

AllSaintsDay wrote:You just know that 1/n (log n)( log log n) is going to show up "fashionably late."

log log n is always less than 8... at least so I hear
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doogly
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### Re: Infinity

It certainly can reach infinity, but it is interesting that any positive power of x will always beat it, regardless of what it is, and x^(-a)ln(x) will go to zero no matter how tiny a positive number a is. The divergence of the logarithm at 0 is also fun - it is so close to behaving, and yet not quite. This makes them interesting in quantum field theory, for example, where the degree of divergence actually matters. We do tricks like subtract infinity from infinity and get an answer like the mass of the electron. There are also a lot of pure math areas where people care about the degree of singularity in a problem; I think some geometers care, but it's mostly analysts, whom I do not so frequently talk to. But I can prove they exist! [Which is about all they can do for me when I ask them questions ]
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mike-l
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### Re: Infinity

doogly wrote:It certainly can reach infinity, but it is interesting that any positive power of x will always beat it, regardless of what it is, and x^(-a)ln(x) will go to zero no matter how tiny a positive number a is. The divergence of the logarithm at 0 is also fun - it is so close to behaving, and yet not quite. This makes them interesting in quantum field theory, for example, where the degree of divergence actually matters. We do tricks like subtract infinity from infinity and get an answer like the mass of the electron. There are also a lot of pure math areas where people care about the degree of singularity in a problem; I think some geometers care, but it's mostly analysts, whom I do not so frequently talk to. But I can prove they exist! [Which is about all they can do for me when I ask them questions ]

Geometers don't believe in e^x or ln x.
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Tirian
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### Re: Infinity

t0rajir0u wrote:Paradoxes only happen if you think of infinity as a number. It's much more satisfying, and much more relevant to modern mathematics, to think of infinity as a place.

I think it's best to stop thinking of infinity as a single thing. There are many of them that actually are numeric in nature, like [imath]\omega[/imath], [imath]c[/imath], and [imath]\aleph_0[/imath]. And then there is [imath]\infty[/imath], which isn't so much a number except where it is formally defined in the theory of complex functions (and even then it isn't really really a number). When one says [imath]lim_{n\rightarrow\infty}ln(x)=\infty[/imath], it's just shorthand for saying that the natural limit grows without bound as x becomes arbitrarily large. It's kind of useful but also kind of unfortunate that it's the same notation that we use for limits involving actual numbers, but you shouldn't infer from that that [imath]\infty[/imath] actually is a number.

kevmus
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### Re: Infinity

The simple (not-completely-mathematically-sound) way to look at it:

Log10(x) rounded down plus one is the number of digits x has.

If you raise x enough, the numbers of digits x has will always keep increasing. If you raise x to a limit of infinity, then x will have infinite digits.

Therefore, logs go to infinity as x goes to infinity.

minno
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### Re: Infinity

The easiest way to think of a limit of [imath]\infty[/imath] at [imath]\infty[/imath] is to think of it as a contest. If someone challenges you to say something like, "I bet you can't make ln(x) bigger than 100", then you can say "anything bigger than e100 works" and win. The limit is [imath]\infty[/imath] at [imath]\infty[/imath] if the second person can always win, no matter what the first person says.
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kaimason1
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### Re: Infinity

Infinity, as many have stated, is merely a concept. It represents no number, although larger numbers tend to work quite well to estimate it in context (in quickly limiting functions, 100 can work fine, while googol may be good for 1/x).
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doogly
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### Re: Infinity

mike-l wrote:
doogly wrote:It certainly can reach infinity, but it is interesting that any positive power of x will always beat it, regardless of what it is, and x^(-a)ln(x) will go to zero no matter how tiny a positive number a is. The divergence of the logarithm at 0 is also fun - it is so close to behaving, and yet not quite. This makes them interesting in quantum field theory, for example, where the degree of divergence actually matters. We do tricks like subtract infinity from infinity and get an answer like the mass of the electron. There are also a lot of pure math areas where people care about the degree of singularity in a problem; I think some geometers care, but it's mostly analysts, whom I do not so frequently talk to. But I can prove they exist! [Which is about all they can do for me when I ask them questions ]

Geometers don't believe in e^x or ln x.

I am not sure why you would say this.
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mike-l
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### Re: Infinity

doogly wrote:
Geometers don't believe in e^x or ln x.

I am not sure why you would say this.

It was a joke, e^x and ln x are not algebraic functions.
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hawkmp4
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### Re: Infinity

minno wrote:The easiest way to think of a limit of [imath]\infty[/imath] at [imath]\infty[/imath] is to think of it as a contest. If someone challenges you to say something like, "I bet you can't make ln(x) bigger than 100", then you can say "anything bigger than e100 works" and win. The limit is [imath]\infty[/imath] at [imath]\infty[/imath] if the second person can always win, no matter what the first person says.

I like this explanation. It's a good way to describe to people the concept of making a function arbitrarily large. Thanks! I'll use this when helping teach a Calc 1 workshop I do for a job.
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doogly
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### Re: Infinity

The exponential map is highly important in any sort of differential geometry. I suppose they are less common in algebraic geometry, but although you can't do as much algebra with non-algebraic functions, you can ask the super interesting questions "what does algebra teach about analysis" and the reverse, which makes it perhaps more interesting to me than is warranted. And I was probably just being unfortunately humorless, but in my defense it was the morning. And I like transcendental curves.
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snowyowl
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### Re: Infinity

Back to the OP, you remind me of what my director of studies told me about a candidate he had interviewed (too see if the candidate was good enough to study at the university). He asked the candidate to sketch the graph of a certain function (which one is not important). The candidate drew it wrong. After trying to find out where the candidate was going wrong, the director of studies asked him to sketch the graph y=e^x. The student drew a reasonably accurate sketch, but with a vertical asymptote at x=99. The supervisor asked why there was an asymptote there. The candidate said "Because my calculator gives an error if I try to work it out".

He didn't get in.
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mike-l
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### Re: Infinity

doogly wrote:The exponential map is highly important in any sort of differential geometry. I suppose they are less common in algebraic geometry, but although you can't do as much algebra with non-algebraic functions, you can ask the super interesting questions "what does algebra teach about analysis" and the reverse, which makes it perhaps more interesting to me than is warranted. And I was probably just being unfortunately humorless, but in my defense it was the morning. And I like transcendental curves.

Yes, it was just a (weak) joke.
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minno
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### Re: Infinity

hawkmp4 wrote:
minno wrote:The easiest way to think of a limit of [imath]\infty[/imath] at [imath]\infty[/imath] is to think of it as a contest. If someone challenges you to say something like, "I bet you can't make ln(x) bigger than 100", then you can say "anything bigger than e100 works" and win. The limit is [imath]\infty[/imath] at [imath]\infty[/imath] if the second person can always win, no matter what the first person says.

I like this explanation. It's a good way to describe to people the concept of making a function arbitrarily large. Thanks! I'll use this when helping teach a Calc 1 workshop I do for a job.

I can't claim credit for this explanation, but I also can't remember who I heard it from. Just spread it on, the more people that know it the better.
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mike-l
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### Re: Infinity

minno wrote:
hawkmp4 wrote:
minno wrote:The easiest way to think of a limit of [imath]\infty[/imath] at [imath]\infty[/imath] is to think of it as a contest. If someone challenges you to say something like, "I bet you can't make ln(x) bigger than 100", then you can say "anything bigger than e100 works" and win. The limit is [imath]\infty[/imath] at [imath]\infty[/imath] if the second person can always win, no matter what the first person says.

I like this explanation. It's a good way to describe to people the concept of making a function arbitrarily large. Thanks! I'll use this when helping teach a Calc 1 workshop I do for a job.

I can't claim credit for this explanation, but I also can't remember who I heard it from. Just spread it on, the more people that know it the better.

I like to explain epsilon-delta proofs in this way as well, I heard it first from a colleague, no clue where he heard it from or if he just made it up, but it tends to help! Existentials and Universals are confusing at first, it's nice whenever you can find a nice way of explaining it.
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AllSaintsDay
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