CSflim

Not just in calculus, but in all algebra. A variable stands for a number, but infinity is not a number.

You can think of it this way:

5/0 = x
Now, if you think it out logically, x becomes infinity. The sameller the denominator gets the larger x gets, so when the denominator is as small as possible, x is as large as possible, hence as the denominator "approaches" zero, x approaches infinity. But we cannot say that x = infinity, instead, x is undefined.
Heres why:
5/0 = x ; x = infinity
6/0 = y ; y = infinity
x=y
therefore 5 = 6.

All numbers are finite. By this I don't mean that there is a finite AMOUNT of numbers, but rather that any particluar number is finite. Hence infity is not a number and you cannot perform artithmetic operations on it.
It is similar in some ways to zero:
X*Z = Y*Z
therefore X=Y
but what if z=0? Well in that case, we can't "cancel" the Zs, as that would be divideing by zero, hence coming up with an undefined term.
In a similar way that algebra and artithmetic seem to work slightly counter-intuitively when zero is concerned, it is the same with infinity, only more-so!
Infinity + Any number still equals infinity! Or is undefined.
Infinity - any number still equals infinitiy or undefined: Inifiny - Infinity = ?
You may straigt away say zero, but think about it. You have infinte marbles, and you take away half of them. You are taking away half of infinity,which is infinity. Hence infinity - infinity = infinity!
Infinity * any number still equals infinity, or undefined: What if we multiply infinity by zero. what is the result? Again undefined!

So, in mathermatics, arithemetic involving infinitiy is very awkward. Infinity is not a number. But does it EXIST?

Well, as already has been stated, it certaintly exists in the world of mathermatics, we have defined, therefore it exists!

But does it exist in the empirical world? Well we certaintly can't MEASURE a length of infinity, even if the universe is infitly big (which I doubt).
Many systems believed to be continous, were proved to be infact quatisised (or discrete) in the 1900s, such as the frequencies of light, and the energies of particles. But if there remains any purely continous system, we can deduce the existence of infinity.
Take for instance space. If you accept that length is a continous dimension and is not discrete, like a computer screen which is made up of discrete pixels.
Take a meter stick. Now somewhere on that meter stick place an imaginary atom. Now move it, and place it some where else? How many possible positions exist for this atom? An infinite number!* Infinity exists.
Now subtract from you meter stick 10cm. Does this change the number of available positions for your atom? No, they are still infinite. What if you break it in two? What if you increase it's length. what if you "square" it? Instead of a one dimensional length, use a square metre area. Still infinity. But surely there are "more" available positions in a whole 1m^2 area than in a 1m length? Well, yes and no...remember that infinity is not a number, and so one infinity is not more than another.

*Two subnotes:
1 The number of positions is not equal to the number of atoms that make up a one meter length. Remember that now that you have placed you atom, you can push it left by half the width of an atom...or a quarter...or an eight...etc.
2 I am aware that an atom does not ocupy a discrete location, I am using it purely as an analogy.

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