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TuringTest

  • 3 years ago

\[\lim_{x\to\infty}x=?\]

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  1. Math4Life
    • 3 years ago
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    What does that mean?

  2. TuringTest
    • 3 years ago
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    are we allowed to say this limit Does not exist? or is that an incorrect statement?

  3. TuringTest
    • 3 years ago
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    must we say the limit exists, and it is \(+\infty\) ?

  4. Math4Life
    • 3 years ago
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    Sorry i'm not that advanced in math yet

  5. TuringTest
    • 3 years ago
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    @FoolForMath @JamesJ @Zarkon @Mr.Math please clear up a technicality @Freckles here is our question

  6. Math4Life
    • 3 years ago
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    @TuringTest what is your question?

  7. freckles
    • 3 years ago
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    The question is can you interpret the limit being infinity as the limit does not exist?

  8. freckles
    • 3 years ago
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    And I'm not saying if the limit does not exist, then the limit is infinity Remember the logic stuff we learned in Discrete math p->q does not imply q->p

  9. freckles
    • 3 years ago
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    I'm saying if the limit is infinity, then the limit does not exist

  10. TuringTest
    • 3 years ago
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    I think the limit does exist, and it is \(=\infty\) freckles says that you can also say the limit does not exist because \(\infty\) is not a number I just want to get some more input

  11. eigenschmeigen
    • 3 years ago
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    i posted this in the other thread, but since we moved here here we go The limit of f(x) as x approaches a is L if and only if, given e > 0, there exists d > 0 such that 0 < |x - a| < d implies that |f(x) - L| < e can we apply this as a formal definition?

  12. TuringTest
    • 3 years ago
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    that would probably help clear up the matter if we can do it right ^

  13. freckles
    • 3 years ago
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    No amistre64 is not allowed to talk. :p

  14. eigenschmeigen
    • 3 years ago
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    if we do accept that as our definition i see a potential problem: |f(x) - L| < e so we want |f(x) - infinity| < e which is a bit weird

  15. amistre64
    • 3 years ago
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    there are more than 1 cardinality of infinity; so i believe its DNE since it cannot have a definitive value

  16. TuringTest
    • 3 years ago
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    hm... good point

  17. amistre64
    • 3 years ago
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    does the limit settle to infinity? if so, which infinity are we discussing :)

  18. TuringTest
    • 3 years ago
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    but still... I am not completely convinced (perhaps I never will be) this seems to undermine the difference between things like\[\lim_{x\to\infty}\sin x\]and the one I posted...

  19. amistre64
    • 3 years ago
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    sin(x) doesnt settle to anything; much less infinity

  20. amistre64
    • 3 years ago
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    if we cant determine the limit that it settles down to; it is undefined

  21. experimentX
    • 3 years ago
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    this is getting interesting

  22. freckles
    • 3 years ago
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    I think DNE can be used whenever the function is oscillating, left limit does not equal right limit, the limit is infinity

  23. amistre64
    • 3 years ago
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    in sin(x) we have a bound; but in x we are boundless is the only diff i see

  24. TuringTest
    • 3 years ago
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    clearly lim sinx to infty DNE oh... Zarkon came online, I beet he can help!

  25. amistre64
    • 3 years ago
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    sin(x) doesnt need to act like x(x) does it?

  26. TuringTest
    • 3 years ago
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    right, so why can we say they both DNE ? that seems to vague to me...

  27. amistre64
    • 3 years ago
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    becasue neither one of them has a point that they settle down to

  28. Zarkon
    • 3 years ago
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    the limit does not exist. you are not using the correct formal definition of a limit (as \(x\to\infty\))

  29. Zarkon
    • 3 years ago
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    you can say that the limit diverges to infinity

  30. TuringTest
    • 3 years ago
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    hm... so "blah, blah diverges to +/- infty" implies that the limit also DNE ?

  31. Zarkon
    • 3 years ago
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    for a limit to converge it has to converge to a number...infinity is not a number

  32. TuringTest
    • 3 years ago
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    Freckles wins! I concede, happy to have learned something :)

  33. experimentX
    • 3 years ago
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    It seems, 0 < |x - a| < d implies that |f(x) - L| < e |x - infinity| < d <----- looks like this 'd' buddy cannot be defined exactly should imply |f(x) - infinity| < e <---- and same goes to 'e' buddy

  34. FoolForMath
    • 3 years ago
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    I would say that the limit of \(f(x)=x\) as x tends to infinity is \(+\infty\) and save the DNE (does not exist) for those cases where left hand limit \(\neq \) right hand limit.

  35. FoolForMath
    • 3 years ago
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    Precisely where there is a hole in the graph or there is no graph at all.

  36. FoolForMath
    • 3 years ago
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    To me undefined and infinity are two different things.

  37. FoolForMath
    • 3 years ago
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    There is a nice discussion on this in M.SE: http://math.stackexchange.com/questions/36289/

  38. TuringTest
    • 3 years ago
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    "To me undefined and infinity are two different things." yes, that's how I felt as well, but apparently if the limit tends to infinity it technically does not exist. I guess just saying that the limit is infinity is a more specific way, or at least a reason for, stating that the limit DNE

  39. TuringTest
    • 3 years ago
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    @FoolForMath wow, that's a very thorough answer from Qiaochu, and actually helped me understand ideas like rings and fields

  40. FoolForMath
    • 3 years ago
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    @TuringTest: Yes, the limit tends to infinity is technically does not exist, I see people often use thee two interchangeably, it's fine because technically \( ∞∉\mathbb{R}\). And I agree, that's a dainty answer :)

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