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my friend and I talking about this, he say equal cus both infinite, i say:
lim x→ inf (floor(x/3) \> floor(x/7))
> *Originally posted by **[UnknownGuardian](/forums/4/topics/309241?page=1#posts6539014):***
>
> [False](http://www.wolframalpha.com/input/?i=lim+x+to+infinity+floor%28x%2F3%29+%3E+lim+x+to+infinity+floor%28x%2F7%29).
edited,
[http://www.wolframalpha.com/input/?i=lim+x+to+infinity+%28floor%28x%2F3%29+%3E+floor%28x%2F7%29%29](http://www.wolframalpha.com/input/?i=lim+x+to+infinity+%28floor%28x%2F3%29+%3E+floor%28x%2F7%29%29)



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[False](http://www.wolframalpha.com/input/?i=lim+x+to+infinity+floor%28x%2F3%29+%3E+lim+x+to+infinity+floor%28x%2F7%29).
**EDIT** x1 for your edit: [infinity x 2 might be less than infinity](http://www.wolframalpha.com/input/?i=2*infinity+%3E+infinity)
**EDIT** x2 [I think you have to separate limits](http://www.wolframalpha.com/input/?i=limit+x+to+infinity+%282x%29+%3E+limit+y+to+infinity+%28y%29), not group em.



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Both those limits are infinity. Not sure what you were going to prove with it either.



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Considering that we [proved that there are the same number of rational numbers as integers](http://www.kongregate.com/forums/4gameprogramming/topics/308591mathproblemnotmyhomework?page=3#posts6532558) your friend is correct. Both sets are the same size.



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Multiples of three can be enumerated as follows: f(x) = 3x, where x is an integer. Multiples of seven can be enumerated similarly: g(x) = 7x, where x is an integer.
Both of these functions are onetoone mappings, and what’s more, the two can be combined: h(x) = 3x/7, where x is a multiple of 7.
h(x) provides a onetoone mapping from the set of integer multiples of 7 onto the set of integer multiples of 3. In other words, every multiple of 7 can be paired with a multiple of 3, and no numbers from either set will be left out.
Therefore, the sets are the same size. Q.E.D.
See also: [Hilbert’s Grand Hotel](http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel).



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> *Originally posted by **[player\_03](/forums/4/topics/309241?page=1#posts6539105):***
>
> Multiples of three can be enumerated as follows: f(x) = 3x, where x is an integer. Multiples of seven can be enumerated similarly: g(x) = 7x, where x is an integer.
>
> Both of these functions are onetoone mappings, and what’s more, the two can be combined: h(x) = 3x/7, where x is a multiple of 7.
>
> h(x) provides a onetoone mapping from the set of integer multiples of 7 onto the set of integer multiples of 3. In other words, every multiple of 7 can be paired with a multiple of 3, and no numbers from either set will be left out.
>
> Therefore, the sets are the same size. Q.E.D.
>
> See also: [Hilbert’s Grand Hotel](http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel).
fascinating :)



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Worth mentioning though that your friend is correct but for the wrong reason: “equal cus both infinite” is wrong. For instance, there are infinitely many integers, and infinitely many real numbers, but there are infinitely _more_ real numbers than integers. The two sets are not in the same class of infinity, and you can prove it.
As for the limit when x tends to infinity of floor(x/3) / floor(x/7), it’s 7/3.



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I liked this one:
[http://www.youtube.com/watch?v=VjZyOTES6iQ](http://www.youtube.com/watch?v=VjZyOTES6iQ)
Oh, and it gets a bit less physics related from 3:18



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What’s `infinity / infinity`?



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> *Originally posted by **[SWATLLAMA](/forums/4/topics/309241?page=1#posts6541260):***
>
> 1.
False.
It’s undefined.
infinity / infinity = 1
→ (infinity + infinity) / infinity = 1
→ (infinity / infinity) + (infinity / infinity) = 1
1+1=1
**ERROR**
[http://www.philforhumanity.com/Infinity\_Divided\_by\_Infinity.html](http://www.philforhumanity.com/Infinity_Divided_by_Infinity.html)



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> *Originally posted by **[ErlendHL](/forums/4/topics/309241?page=1#posts6541204):***
>
> What’s `infinity / infinity`?
This is one of the cases where we can’t really say anything.
Other such cases include:
0\*inf, 1^inf, inf – inf



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> *Originally posted by **[someone93](/forums/4/topics/309241?page=1#posts6541700):***
> > *Originally posted by **[ErlendHL](/forums/4/topics/309241?page=1#posts6541204):***
> >
> > What’s `infinity / infinity`?
>
> This is one of the cases where we can’t really say anything.
>
> Other such cases include:
> 0\*inf, 1^inf, inf – inf
can you explain why 1 ^ infinity is obscure? oo



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I can’t remember specifics. But we showed it in my Uni. intro math.



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> *Originally posted by **[someone93](/forums/4/topics/309241?page=1#posts6541755):***
>
> I can’t remember specifics. But we showed it in my Uni. intro math.> *Originally posted by **[qwerberism](/forums/4/topics/309241?page=1#posts6541709):***
> > *Originally posted by **[someone93](/forums/4/topics/309241?page=1#posts6541700):***
> > > *Originally posted by **[ErlendHL](/forums/4/topics/309241?page=1#posts6541204):***
> > >
> > > What’s `infinity / infinity`?
> >
> > This is one of the cases where we can’t really say anything.
> >
> > Other such cases include:
> > 0\*inf, 1^inf, inf – inf
>
> can you explain why 1 ^ infinity is obscure? oo
I think it’s similar to why 0\*inf is obscure. 0 is similar to 1/infinity. so 0\*inf is similar to infinity/infinity, which is obscure.
1^inf = (1+0)^inf = (10)^inf which is similar to (11/infinity)^infinity and (1+1/infinity)^infinity, which are obscure. Basically if you were to analyze infinity as some number x approaching infinity, then 1 is really some number y approaching 1. Then y^x isn’t necessarily 1 and is obscure.



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> *Originally posted by **[qwerberism](/forums/4/topics/309241?page=1#posts6541709):***
> > *Originally posted by **[someone93](/forums/4/topics/309241?page=1#posts6541700):***
> > > *Originally posted by **[ErlendHL](/forums/4/topics/309241?page=1#posts6541204):***
> > >
> > > What’s `infinity / infinity`?
> >
> > This is one of the cases where we can’t really say anything.
> >
> > Other such cases include:
> > 0\*inf, 1^inf, inf – inf
>
> can you explain why 1 ^ infinity is obscure? oo
Let’s say 1^inf = x
ln(1^inf) = ln(x)
inf\*ln(1) = ln(x)
inf\*0 = ln(x)



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`1 ^ infinity` should `= 1` I think??
* * *
`
3/0 = inf
inf * 0 = 3
`
100% legit.



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`Proof:
log b to the base a = (log b)/(log a)
Example:
log 8 base 2 = 3  2^3 = 8
(log 8)/(log 2) = 3
log 8 base 1
= (log 8)/(log 1)
= (log 8)/0 = infinity
It follows that 1^infinity = 8
Similarly
log 9 base 1 = infinity
It follows that 1^infinity = 9
1^infinity can have any positive value greater than or equal to 1.
1^infinity >= 1
1^(infinity) = 1/1^(infinity)
0 <= 1^(infinity) <= 1
1^infinity is indeterminate (undefined).`



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x/0 is not infinity, it’s indeterminate and contextual.



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> *Originally posted by **[BigJM](/forums/4/topics/309241?page=1#posts6543629):***
>
> x/0 is not infinity, it’s indeterminate and contextual.
x/0 = a (or a\*0 = x), for x != 0, is undefined, not indeterminate. There is **not a single** solution.
0/0 = a (or a\*0 = 0) on the other hand is indeterminate. There is **not a unique** solution (it’s true for every value of a).



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It is unable to be determined; therefore, it is indeterminate.



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> *Originally posted by **[BigJM](/forums/4/topics/309241?page=1#posts6546560):***
>
> It is unable to be determined; therefore, it is indeterminate.
You would be correct.
> An indeterminate system is a system of simultaneous equations (especially linear equations) which has **infinitely many solutions** or **no solutions at all.** The system may be said to be underspecified.



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[Complex Infinity?](http://www.wolframalpha.com/input/?i=x%2F0)



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I think it’s assuming that x is in the extended complex plane. That’s why I said it’s meaning is contextual.



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> *Originally posted by **[BigJM](/forums/4/topics/309241?page=1#posts6546873):***
>
> I think it’s assuming that x is in the extended complex plane. That’s why I said it’s meaning is contextual.
Oddly I got the same result when I try `Re[x]/0`
Ran it through Mathematica 8:
`
In[1]:= Re[x]/0
During evaluation of In[1]:= Power::infy: Infinite expression 1/0 encountered. >>
Out[1]= ComplexInfinity
`
