## TuringTest Group Title $\lim_{x\to\infty}x=?$ 2 years ago 2 years ago

1. Math4Life Group Title

What does that mean?

2. TuringTest Group Title

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

3. TuringTest Group Title

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

4. Math4Life Group Title

Sorry i'm not that advanced in math yet

5. TuringTest Group Title

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

6. Math4Life Group Title

7. freckles Group Title

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

8. freckles Group Title

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 Group Title

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

10. TuringTest Group Title

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 Group Title

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 Group Title

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

13. freckles Group Title

No amistre64 is not allowed to talk. :p

14. eigenschmeigen Group Title

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 Group Title

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

16. TuringTest Group Title

hm... good point

17. amistre64 Group Title

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

18. TuringTest Group Title

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 Group Title

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

20. amistre64 Group Title

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

21. experimentX Group Title

this is getting interesting

22. freckles Group Title

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

23. amistre64 Group Title

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

24. TuringTest Group Title

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

25. amistre64 Group Title

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

26. TuringTest Group Title

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

27. amistre64 Group Title

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

28. Zarkon Group Title

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

29. Zarkon Group Title

you can say that the limit diverges to infinity

30. TuringTest Group Title

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

31. Zarkon Group Title

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

32. TuringTest Group Title

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

33. experimentX Group Title

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 Group Title

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 Group Title

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

36. FoolForMath Group Title

To me undefined and infinity are two different things.

37. FoolForMath Group Title

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

38. TuringTest Group Title

"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. experimentX Group Title
40. TuringTest Group Title

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

41. FoolForMath Group Title

@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 :)