How do you "approach" infinity?

[j0k3r]

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

[Yoni]

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

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.

[j0k3r]

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

[Yoni]

(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.