TuringTest
  • TuringTest
\[\lim_{x\to\infty}x=?\]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
What does that mean?
TuringTest
  • TuringTest
are we allowed to say this limit Does not exist? or is that an incorrect statement?
TuringTest
  • TuringTest
must we say the limit exists, and it is \(+\infty\) ?

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anonymous
  • anonymous
Sorry i'm not that advanced in math yet
TuringTest
  • TuringTest
@FoolForMath @JamesJ @Zarkon @Mr.Math please clear up a technicality @Freckles here is our question
anonymous
  • anonymous
@TuringTest what is your question?
freckles
  • freckles
The question is can you interpret the limit being infinity as the limit does not exist?
freckles
  • freckles
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
freckles
  • freckles
I'm saying if the limit is infinity, then the limit does not exist
TuringTest
  • TuringTest
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
anonymous
  • anonymous
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?
TuringTest
  • TuringTest
that would probably help clear up the matter if we can do it right ^
freckles
  • freckles
No amistre64 is not allowed to talk. :p
anonymous
  • anonymous
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
amistre64
  • amistre64
there are more than 1 cardinality of infinity; so i believe its DNE since it cannot have a definitive value
TuringTest
  • TuringTest
hm... good point
amistre64
  • amistre64
does the limit settle to infinity? if so, which infinity are we discussing :)
TuringTest
  • TuringTest
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...
amistre64
  • amistre64
sin(x) doesnt settle to anything; much less infinity
amistre64
  • amistre64
if we cant determine the limit that it settles down to; it is undefined
experimentX
  • experimentX
this is getting interesting
freckles
  • freckles
I think DNE can be used whenever the function is oscillating, left limit does not equal right limit, the limit is infinity
amistre64
  • amistre64
in sin(x) we have a bound; but in x we are boundless is the only diff i see
TuringTest
  • TuringTest
clearly lim sinx to infty DNE oh... Zarkon came online, I beet he can help!
amistre64
  • amistre64
sin(x) doesnt need to act like x(x) does it?
TuringTest
  • TuringTest
right, so why can we say they both DNE ? that seems to vague to me...
amistre64
  • amistre64
becasue neither one of them has a point that they settle down to
Zarkon
  • Zarkon
the limit does not exist. you are not using the correct formal definition of a limit (as \(x\to\infty\))
Zarkon
  • Zarkon
you can say that the limit diverges to infinity
TuringTest
  • TuringTest
hm... so "blah, blah diverges to +/- infty" implies that the limit also DNE ?
Zarkon
  • Zarkon
for a limit to converge it has to converge to a number...infinity is not a number
TuringTest
  • TuringTest
Freckles wins! I concede, happy to have learned something :)
experimentX
  • experimentX
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
anonymous
  • anonymous
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.
anonymous
  • anonymous
Precisely where there is a hole in the graph or there is no graph at all.
anonymous
  • anonymous
To me undefined and infinity are two different things.
anonymous
  • anonymous
There is a nice discussion on this in M.SE: http://math.stackexchange.com/questions/36289/
TuringTest
  • TuringTest
"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
experimentX
  • experimentX
http://www.wolframalpha.com/input/?i=lim+x-%3E0+1%2Fx http://www.wolframalpha.com/input/?i=lim+x-%3Einf+x
TuringTest
  • TuringTest
@FoolForMath wow, that's a very thorough answer from Qiaochu, and actually helped me understand ideas like rings and fields
anonymous
  • anonymous
@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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