It can be "defined" as you see fit for a particular application. It can be infinity, or N/A, or raise an error. There's no one true value for 1/0, or for that matter Tan 90°, that will reveal secrets of the universe.

These are interesting:
1/x, x -> 0+ = +inf
1/x, x -> 0- = -inf
Tan x, x -> 90+° = +inf
Tan x, x -> 90-° = -inf

(1) and (3) are equivalent, and so are (2) and (4). However, it is not obvious, and it does not follow from the fact that they are the same "plus" of inf. Infinities differ, and so do zeroes. inf is not a mathematical value, it's a tag.

For example, here's a mathematical statement: A = (1 + 2 * 3 - 4)^5. It can be rewritten as A = 243 without loss of information concerning A. If you need to calculate P = A*A - 2A + 1/A, you can just replace A with 243.

Now, if B = 1/x, x->0+, you can say that B equals +inf, but if a happy-go-lucky libarts student goes all philosophical on the poor hyperbole and erases the formula, how will he go about calculating Q = B*sin(x), x->0+?
uhh let's see Q = +inf * sin(0+) = +inf * 0+ = lolwut

While, in reality, Q equals 1.

TL;DR: sideways 8 is not a value, it is a tag on an expression that means "this expression does not have a finite value, do not attempt to calculate". So when you have
M = 1 + 1/2! + 1/3! + 1/4! + 1/5! etc,
N = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 etc,
and
T = 1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 etc
you apply some theorems and see that M has a finite value and (more calculations) equals 1.718281828459045blahblah, while N and T are infinite and should be left as they are.

Infinities are not equal, and "zeroes" are not equal. For example, radioactive decay is exponential; it's one of the most "powerful" individual functions (although of course you can take an exponent of an exponent). So, while lim exp(-1/x) at x->+0 is zero, and lim 1/x at x->+0 is +inf, lim exp(-1/x)/x at x->+0 is still zero.

Abstract algebra, a higher level of abstraction than numbers. With numbers, you can count apples, oranges, cars, houses, shades of the color purple, etc. Quantities share the same properties no matter what you count, which is why math is useful. Now, abstract algebra deals with similarities between numbers, vectors, matrices, functions, and rotations of a Rubik cube - these are called structures.

A "field" is a certain kind of structure. The exact definition of a field is rather complicated, but these are among the requirements:
It can be shown that real numbers are a field. If you define infinity and add it to real numbers, the resulting new structure will no longer be a field, which means that theorems which have been proven to work in fields might not work in the new structure you just defined.

Like all mathematical functions, the square root is a tool.

As an engineer, I gather up all of my tools to solve problems. Sometimes things get tricky and things like square roots can get a job done.

When you don't solve for a decimal value of a square root, sometimes you find that it'll cancel out with something, or add up into something nice and tidy.

You can use i (or j [as mentioned earlier to be the sqrt(-1)]) to pull a line into two dimensions. That has all kinds of engineering uses.

As for precision, you can look at it from many points of view. As a mathematician, cosmologist, carpenter, electrical engineer, mechanical engineer and chemical engineer, you'll look it completely different. Some may use more significant digits in the decimal value. Others may use no decimal at all and keep it in its pure irrational form.

So, it's purpose is to be a tool to get things done. Once you know how it works, you can use it as a tool. Some people use it little, some people a lot. You might use it to figure out how large to cut something. And then again, you might never use it at all.

In order to make myself a bit clearer, I will talk specifically about sqrt(2).

The notation "sqrt(2)" is indeed a replacement for a specific number. Which one? Well, that guy which, when squared, gives you 2 back (which is the DEFINITION of square root). If you try to write it in decimal notation, you kind of get 1,4142136..., with no end to the string of digits, nor clear repetition pattern. So the notation for the square root is immensely helpful when expressing this number -- if you try to do it in decimal notation, you either get an infnite, hard-to-describe sequence of digits OR you truncate this sequence somewhere and then get only approximations. So in a sense you could claim the ONLY way to write the square root of 2 is, well, using the concept of square root.

Now, I think I know the ACTUAL question you want to ask:

"What is a real number?"

I mean, it is not too hard to construct the natural numbers, and from them the integers, and from these the rational numbers. However, it is not hard to show that if sqrt(2) exists, it cannot be a rational number. So if we WANT sqrt(2) to exist, we somehow need to construct a bigger set of numbers. And so we create this set R called "real numbers", which happens to contain sqrt(2), pi ,e and many other useful numbers.

Now, the construction of R is fairly technical, and that is why most people do not see it in their usual lives -- the analogy of R with points in a real line, though highly imprecise, is enough to get people to work with R. Even many Real Analysis books do not bother with the actual construction; instead they just bother to mention the axioms which ultimately define what R is. Hey, I can tell you what they are in 1 line:

"R is a complete ordered field."

Vaguely:
-- "Field" means that you can define the 4 basic operations, with their usual properties.
-- "Ordered" means that for any x,y in R you have x<y, x=y or x>y, with the usual properties.
-- "Complete" is the trickiest one. It says that any subset of real numbers which has some upper bound must have a SUPREME, which is a SMALLEST upper bound. More or less, this means that R has no "holes";

Amazingly enough, these 3 properties are enough to obtain all we know about R, including all Calculus -- and the existence of square roots for positive numbers. In a sense, R is the ONLY set which satisfies them.

Cheers!

I don't understand what you mean by "why the square root exists". It is a mathematical contruct. The real numbers contain all square roots of non-negative real numbers by construction. It is this construction process (and hence the real numbers themselves) that has to be understood. You usually start by properly defining the natural numbers, then integers, then quotients (of integer numbers, no zero in the denominator). I think so far everyone has a pretty good intuitive understanding. If, at this point, you look closely at the rational numbers, you see that there are "holes" in the seemingly continuous line of rational numbers, meaning: there are certain values that can be proven to not be rational (like sqrt(2). In fact, those holes are "almost everywhere"), but they can be approximated arbitrarily close by rational numbers - after all, we can make the denominator arbitrarily large, so the rational numbers are dense. For many problems, it would be better to deal with a field of numbers without such holes, so what you do, basically, is say "Ok, why don't we take the rational numbers and add all of those holes as extra numbers?".
So now we have defined a bigger field of numbers, the real numbers, and square roots are real numbers by construction. Unfortunately, we do not have an easy way of writing "most" real numbers, and they are more complicated in some regards, but we have gotten rid of the "holes" that plagued the rational numbers.

As Starmaker explained: unless you know what you are doing and what exactly you are writing down, you should avoid infinity. You should usually state the exact limiting process you are describing (in this case, you are taking the left-side limit of tan at 90°, but we haven't really defined what exactly this limiting process is either). In certain circumstances, using the extended real (or complex) numbers is useful, but then you first have to take care of well-definedness. In particular, if expressions such as "(+infinity) * (-infinity)" or "infinity/infinity" occur, it is hard to make any sense of them.


edit: damn it, ninja'd

My apologies for being rude.
For me the square root is just a mathematical operation, nothing more. I didn't realized there could be philosophy involved so that's why I said the question is childish. That was my first impression. Now I'll step back and let the philosophers speak. ;)

Recent posts have been very unphilosophical, sorry.

Thank you everyone for the help. You made it very clear to me and I really appreciate it. Thanks again.