## TuringTest 3 years ago $\lim_{x\to\infty}x=?$

1. Math4Life

What does that mean?

2. TuringTest

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

3. TuringTest

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

4. Math4Life

Sorry i'm not that advanced in math yet

5. TuringTest

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

6. Math4Life

7. freckles

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

8. 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

9. freckles

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

10. 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

11. eigenschmeigen

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

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

13. freckles

No amistre64 is not allowed to talk. :p

14. eigenschmeigen

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

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

16. TuringTest

hm... good point

17. amistre64

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

18. 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...

19. amistre64

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

20. amistre64

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

21. experimentX

this is getting interesting

22. freckles

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

23. amistre64

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

24. TuringTest

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

25. amistre64

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

26. TuringTest

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

27. amistre64

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

28. Zarkon

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

29. Zarkon

you can say that the limit diverges to infinity

30. TuringTest

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

31. Zarkon

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

32. TuringTest

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

33. 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

34. FoolForMath

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

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

36. FoolForMath

To me undefined and infinity are two different things.

37. FoolForMath

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

38. 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

39. experimentX
40. TuringTest

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

41. FoolForMath

@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 :)