One of the fundamental branches of modern math – differential and integral calculus – is based on the concept of *limits*. In some ways, limits are a very intuitive concept – but the formalism of limits can be extremely confusing to many people.

Limits are basically a tool that allows us to get a handle on certain kinds

of equations or series that involve some kind of infinity, or some kind of value that is *almost* defined. The informal idea is very simple; the formalism is *also* pretty simple, but it’s often obscured by so much jargon that it’s hard to relate it to the intuition.

The use of limits for finding *almost* defined values sounds tricky, but

it’s really pretty simple, and it makes for a very good illustration.

Think of the simple function: f(x)=(x-1)/(sqrt(x)-1). Just looking at it, you should be able to quickly see that its value at x=1 is undefined – f(1)=(1-1)/(1-1)=0/0.

But let’s look at what happens as we get *close to* x=1.

f(2)=2.414. f(1.5)=2.22. f(1.2)=2.09. f(1.1)=2.05. f(1.01)=2.005… As we look at values of numbers greater than 1, but ever closer and closer to x=1, we can see that as x gets closer to 1, f(x) gets closer and closer to 2.

The same thing happens from the other direction. f(0.5)=1.71. f(0.9)=1.95. f(0.99)=1.995….

At exactly x=1, the value of the function is undefined – it’s a division by zero. But from either side of 1 – greater or lesser – the closer we get to x=1, the closer f(x) gets to 2. So we say that the *limit* of f(x) as x approaches 1 = 2 – more traditionally written: “lim_{x→1}f(x)=2″.

A simple example of managing infinity with limits is the equation f(x) = (1/x)+4. As x gets larger, f(x) obviously gets closer and closer to 4. It never actually *reaches* four – but it gets closer and closer. For any number ε, no matter how small, you can find *some* value x so that f(x)<4+ε, and after that x, f(x) will *always* be less than 4+ε. Epsilon can be 10^{-800} – and there’s someplace where after some value x, f(x) is always less than 4+10^{-800}.

Which brings us at last to the formal definition of a limit. We’ll start with

the case where we get *infinitely close* to a real value. Given a real-valued function f(x), defined in an open interval *around* a value p (but not necessarily *at* P). Then lim_{x→p}f(x) (the limit of f(x) as x approaches p) = L if and only if for all ε>0, there exists some value δ>0 such that for all x where 0<|x-p|<δ, |f(x)-L|<ε.

That’s really just restating what we did with the example. It’s just a formal way of saying that as x gets closer and closer to p, f(x) gets closer and closer to L. The ε and δ are names for a pair of decreasing values – as x gets closer to p, both ε and δ get closer and closer to 0.

For dealing with infinity, as in our second example, the formal definition is:

Given a real-valued function f, lim_{x→∞}f(x)=L if and only if for all ε>0, there exists some real number n such that for all x>n |f(x)-L|<ε.

This is exactly the same kind of trick that we used in our example of a limit as x approaches infinity – no matter how small a value you pick for epsilon, there is *some* point on the curve after which f(x) *will never* be farther than ε away from L.