[FeLiXe]

you can check out Wikipedia for limits http://en.wikipedia.org/wiki/Limit_%28mathematics%29

[pantone159]

The limit is the number that you get closer and closer to, as you advance in your sequence/sum/whatever.
Formally, I think it is defined something like this:  Pick any small number (often named epsilon) that you like.  If an infinite sum approaches a limit L, that at some point the finite sum gets within epsilon of L (and doesn't leave).  This works no matter how small epsilon is.

Although a finite sum might never reach exactly the limit L, it will get arbitrary close.  Typically, the limit is never reached exactly (without infinitely many steps).  It isn't against the rules for something to get exactly to the limit (e.g. Borek's example), it just usually doesn't happen that way.

BTW, one the most important uses of limits is calculus.  Without the concept of limit, calculus makes no sense.  For example, the derivative of a function is defined as a limit.  Take the ratio ((f(x+dx) - f(x)) / (dx) and find the limit as dx->0.  That is the derivative.  dx never actually reaches 0, the ratio doesn't work then (0/0), it only gets very close.  Limits have exact values just like derivatives do.

[Borek]

Good question. I can imagine sequences that reach the limit (lim x for x -> 1 is just 1 and it reaches the limit) and sequences that don't reach the limit (lim 1/x for x -> ? gets as close to 0 as possible, but never reaches the number). Limit definition doesn't state anything about the limit being reachable or not. But don't rely on me when it comes to math.

On the second thought - limit is reached for every continuous function, so it happens quite often.

[Donaldson Tan]

Jeese. I never thought this would get so much attention.

Haha..

x = 1, y = 0.999...

(x + y)/2 > x
1.999... > 2

(x+y)/2 < x
1.999... < 2

there is really no such number.

Since "<" and ">" are not applicable, what about "=" ?

[xiankai]

Since "<" and ">" are not applicable, what about "=" ?

the inequality signs were there to posit the existence of a number between 0.999... and 1, which does not exist as has been shown. replacing them with '=' does not really prove anything since it could be replaced by '≠' too.

mathematically 0.999... = 1, but physically i refuse to accept it. 

[Inestyne]

0.999... = 0.9 + 0.09 + 0.009 + ...
               = 9(0.1 + 0.01 + 0.001 + ...)
               = 9(-1 + 1 + 0.1 + 0.01 + 0.001 + ...)
               = 9(-1 + ?^?_n=0(1/10)ⁿ)
Since ?^?_n=0rⁿ = 1 / (1 - r) for |r| < 1,
0.999... = 9(-1 + 1/(1 - 1/10))
               = 9(-1 + 10/9)
               = 9(1/9)
               = 1
qed

What? elaborate

[Yggdrasil]

0.999... is basically the sum of a geometric series (http://en.wikipedia.org/wiki/Geometric_series) and can be written in the form:

9 (-1 + Σ_n=0^∞ (1/10)ⁿ)

Since the sum of a geometric series converges to a finite number, we can use this fact to prove that 0.999... = 1.

[Inestyne]

0.999... is basically the sum of a geometric series (http://en.wikipedia.org/wiki/Geometric_series) and can be written in the form:

9 (-1 + Σ_n=0^∞ (1/10)ⁿ)

Since the sum of a geometric series converges to a finite number, we can use this fact to prove that 0.999... = 1.

bingo

[enahs]

While mathematicians were busy arguing over if 0.99999999.... =1  is true or false, chemist and physicists invented all kinds of cool and useful stuff.

Chemist and Physicists >  Mathematicians

What do you think about that relationship?

[Yggdrasil]

And then biologists stole the tools from chemists and physicists to do more cool stuff, get NIH funding, and win chemistry nobel prizes

biologists > chemists and physicists

  (j/k)

[Borek]

When mathematicians proved that 0.99999... = 1 chemists and physicians were unscrupulously using this results in their calculations; they never admitted where did the knowledge came from.

Biologists did not understand the proof nor its significance, so they happily ignored it

mathematicians > chemists + physicists + biologists

 

[pantone159]

It is less true for chemists and biologists, but physicists would have accomplished hardly anything without well-developed mathematics.
 

[Yggdrasil]

And chemists would have accomplished hardly anything without well developed physics and biologists would have accomplished hardly anything without well developed chemistry.  Although the latter is probably more true than the former.

[Maz]

lol, well I guess I'll have to chime in here.

Try to remember, chemistry is simply a fringe branch of physics (q.m. mixed with stat/thermal physics).  Biology is a fringe branch of chemistry.  Mathematics is worthless without relating to the physical world.  Enter Galileo, Kepler, Newton....Most early "physicists" were mathematicians by training who decided to stop futzing about with the useless abstract and started working in the useful abstract. 

So the point is mathematicians ~ physicists > chemists > biologists > social "scientists" (bah)

 

[Mitch]

Mitch > Your Mamma