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: “limx→1f(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 limx→pf(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, limx→∞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.