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