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TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8are we allowed to say this limit Does not exist? or is that an incorrect statement?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8must we say the limit exists, and it is \(+\infty\) ?

Math4Life
 2 years ago
Best ResponseYou've already chosen the best response.0Sorry i'm not that advanced in math yet

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8@FoolForMath @JamesJ @Zarkon @Mr.Math please clear up a technicality @Freckles here is our question

Math4Life
 2 years ago
Best ResponseYou've already chosen the best response.0@TuringTest what is your question?

freckles
 2 years ago
Best ResponseYou've already chosen the best response.1The question is can you interpret the limit being infinity as the limit does not exist?

freckles
 2 years ago
Best ResponseYou've already chosen the best response.1And 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
 2 years ago
Best ResponseYou've already chosen the best response.1I'm saying if the limit is infinity, then the limit does not exist

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8I 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

eigenschmeigen
 2 years ago
Best ResponseYou've already chosen the best response.0i 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
 2 years ago
Best ResponseYou've already chosen the best response.8that would probably help clear up the matter if we can do it right ^

freckles
 2 years ago
Best ResponseYou've already chosen the best response.1No amistre64 is not allowed to talk. :p

eigenschmeigen
 2 years ago
Best ResponseYou've already chosen the best response.0if 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
 2 years ago
Best ResponseYou've already chosen the best response.0there are more than 1 cardinality of infinity; so i believe its DNE since it cannot have a definitive value

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0does the limit settle to infinity? if so, which infinity are we discussing :)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8but 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
 2 years ago
Best ResponseYou've already chosen the best response.0sin(x) doesnt settle to anything; much less infinity

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0if we cant determine the limit that it settles down to; it is undefined

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0this is getting interesting

freckles
 2 years ago
Best ResponseYou've already chosen the best response.1I think DNE can be used whenever the function is oscillating, left limit does not equal right limit, the limit is infinity

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0in sin(x) we have a bound; but in x we are boundless is the only diff i see

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8clearly lim sinx to infty DNE oh... Zarkon came online, I beet he can help!

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0sin(x) doesnt need to act like x(x) does it?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8right, so why can we say they both DNE ? that seems to vague to me...

amistre64
 2 years ago
Best ResponseYou've already chosen the best response.0becasue neither one of them has a point that they settle down to

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.2the limit does not exist. you are not using the correct formal definition of a limit (as \(x\to\infty\))

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.2you can say that the limit diverges to infinity

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8hm... so "blah, blah diverges to +/ infty" implies that the limit also DNE ?

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.2for a limit to converge it has to converge to a number...infinity is not a number

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8Freckles wins! I concede, happy to have learned something :)

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.0It 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

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0I 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.

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0Precisely where there is a hole in the graph or there is no graph at all.

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0To me undefined and infinity are two different things.

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0There is a nice discussion on this in M.SE: http://math.stackexchange.com/questions/36289/

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8"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
 2 years ago
Best ResponseYou've already chosen the best response.0http://www.wolframalpha.com/input/?i=lim+x%3E0+1%2Fx http://www.wolframalpha.com/input/?i=lim+x%3Einf+x

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.8@FoolForMath wow, that's a very thorough answer from Qiaochu, and actually helped me understand ideas like rings and fields

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0@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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