Question

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

Answer 1

What does that mean?

Answer 2

are we allowed to say this limit Does not exist? or is that an incorrect statement?

Answer 3

must we say the limit exists, and it is \(+\infty\) ?

Answer 4

Sorry i'm not that advanced in math yet

Answer 5

@FoolForMath @JamesJ @Zarkon @Mr.Math please clear up a technicality
@Freckles here is our question

Answer 6

@TuringTest what is your question?

Answer 7

The question is can you interpret the limit being infinity as the limit does not exist?

Answer 8

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

Answer 9

I'm saying if the limit is infinity, then the limit does not exist

Answer 10

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

Answer 11

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?

Answer 12

that would probably help clear up the matter if we can do it right ^

Answer 13

No amistre64 is not allowed to talk. :p

Answer 14

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

Answer 15

there are more than 1 cardinality of infinity; so i believe its DNE since it cannot have a definitive value

Answer 16

hm... good point

Answer 17

does the limit settle to infinity? if so, which infinity are we discussing :)

Answer 18

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...

Answer 19

sin(x) doesnt settle to anything; much less infinity

Answer 20

if we cant determine the limit that it settles down to; it is undefined

Answer 21

this is getting interesting

Answer 22

I think DNE can be used whenever the function is oscillating, left limit does not equal right limit, the limit is infinity

Answer 23

in sin(x) we have a bound; but in x we are boundless is the only diff i see

Answer 24

clearly lim sinx to infty DNE
oh... Zarkon came online, I beet he can help!

Answer 25

sin(x) doesnt need to act like x(x) does it?

Answer 26

right, so why can we say they both DNE ?
that seems to vague to me...

Answer 27

becasue neither one of them has a point that they settle down to

Answer 28

the limit does not exist.

you are not using the correct formal definition of a limit (as \(x\to\infty\))

Answer 29

you can say that the limit diverges to infinity

Answer 30

hm...
so "blah, blah diverges to +/- infty" implies that the limit also DNE ?

Answer 31

for a limit to converge it has to converge to a number...infinity is not a number

Answer 32

Freckles wins!
I concede, happy to have learned something :)

Answer 33

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

Answer 34

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.

Answer 35

Precisely where there is a hole in the graph or there is no graph at all.

Answer 36

To me undefined and infinity are two different things.

Answer 37

There is a nice discussion on this in M.SE: http://math.stackexchange.com/questions/36289/

Answer 38

"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

Answer 39

http://www.wolframalpha.com/input/?i=lim+x-%3E0+1%2Fx
http://www.wolframalpha.com/input/?i=lim+x-%3Einf+x

Answer 40

@FoolForMath wow, that's a very thorough answer from Qiaochu, and actually helped me understand ideas like rings and fields

Answer 41

@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 :)