How do you approach infinity if there is no given amount.

[quote author=j0k3r link=board=36;threadid=3886;start=0#msg32108 date=1069972391]
How do you approach infinity if there is no given amount.
[/quote]It's called calculus, and this is the perfect forum to ask questions about it :)
What do you mean by "there's no given amount"? It's n, and n approaches infinity.

Infinity is inifinity, so how do you approach it? Infinity has no given value is what I meant.

(Note to clear confusion: This thread was split from the "Algorithms" thread.)

Approaching infinity is a basic concept in calculus.
Consider these examples to see if you get it without much more explanation:



f(x) = 1 / x

As x becomes bigger and bigger, f(x) becomes smaller and smaller. It can be proven mathematically that you can get as close to 0 as you wish by picking a big enough x. Because of this, f(x) is said to "approach 0, or have a limit of 0, as x approaches infinity". It is written like this:

lim f(x) = 0
x->infinity



f(x) = x^2 - 5

As x becomes bigger, f(x) becomes even bigger. It can be proven that you can get f(x) as large as you want it to be by choosing x correctly. Now, f(x) is said to "approach infinity as x approaches infinity".



f(x) = (3x^3 + 8x^2) / (2x^3 - 5x + 7)

This time, as x becomes bigger, f(x) comes close to 1.5. Like in the first example, you can get it as close as you wish by picking x correctly. f(x) has a limit of 1.5 as x approaches infinity.



f(x) = sin(x)

If you've learned trigonometric functions, you know that this function oscillates periodically between 1 and -1, and never "stops" anywhere. You can't get it to always be near any value you want by picking a large enough x. f(x) has no limit as x approaches infinity.



I can't think of other interesting examples right now, ask more if you still don't understand it.

Here are a few formal definitions:
(These are variants of the Epsilon-Delta definition which Weierstrass liked probably as much as I.)

1. f(x) is said to approach a limit of L as x approaches infinity -
- if and only if -
- for every Epsilon>0, there exists an x0 in R, so that for any x such that x > x0, then |f(x) - L| < Epsilon.

Semi-translation to English:
If you choose some very small positive number (Epsilon), and no matter how small of a number you chose, I can find some point (x0) so that from that point on ("for any x such that x > x0") the function is less than Epsilon away from L ("|f(x) - L| < Epsilon"), then f(x) is said to approach L as x approaches infinity.

2. f(x) is said to approach infinity as x approaches infinity -
- if and only if -
- for every L > 0, there exists an x0 in R, so that for any x such that x > x0, then f(x) > L.

Semi-translation to English:
If you choose some very large number (L), and no matter how large it is, I can find some point (x0) so that from that point on ("for any x such that x > x0") the function is larger than this number, then f(x) is said to approach infinity as x approaches infinity.