I have written several articles which use the concept of limits,
without ever defining "limit." It is about time.Return to my Mathematics pages
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© Copyright 2000, Jim Loy
I have written several articles which use the concept of limits,
without ever defining "limit." It is about time.
Let's take a simple example; let's graph y=x^2/x (using x^2 for x squared). We can simplify this equation; but let's not. We see the graph on the left. As you can see, I drew a question mark at the origin (0,0). The reason for that is that when x=0, y=0/0, which is undefined, or illegal (See Two Equals One?). Well, we can simplify the above equation to y=x. When x=0, then y=0. So, when x=0, maybe x^2/x is also 0. Don't bet on it. It is still undefined. y=x^2/x is the same as y=x everywhere except at that one value of x. There it is undefined. You cannot divide by zero. But for all practical purposes the two equations are the same, and when x=0, y=0. We pretend that x^2/x is 0 when x is 0. But we are careful to never say that it IS zero. We need different notation for cases like this (since we are not really dividing by zero). Here is the notation for this example:
In words, this says "the limit as x approaches zero, of x
squared divided by x equals zero." This is called limit notation, and it
can be defined very precisely. Most of the time, we can define it casually.
First let me mention functions, which I will cover better in a future article.
Functions (like f(x)) are just those things that we graph, like x^2/x or 3x+1
or sin x. Anyway, I'll define a limit like this:
Let x get closer and closer to a. Then if f(x) gets closer and closer to some value, then that value is the limit of f(x). Note that x never reaches a, it just gets closer and closer.
In our example above, as x gets closer and closer to 0, then x^2/x gets closer and closer to 0. So 0 is the limit. We knew that already. But we had to be careful not to divide by zero. We are not concerned with 0/0 at all, just closer and closer to 0/0. We deal with 0/0 by never dividing by zero. We use limits to avoid that trap.
In order to be more careful, we also have to notice that our limit is the same when x gets closer to zero from above or below. Our function of x would not be nearly as useful if these two limits were different.
We also deal with infinity in the same way. On the
right are two cases. In the first, 1/x blows up when x=0. So we use a limit
instead. We don't divide by zero. And we find that the limit is a positive
infinity when x gets close to zero from above, and a negative infinity when x
gets close to zero from below. That function is really in bad shape at zero.
The limits are infinite, and they have different values. The second equation
shows that 1/x goes to zero (gets closer and closer to zero) as x goes to
infinity (grows without bound).
Also remember that infinity is not a number, in the usual sense of the word (See Infinity). It does not follow the rules of arithmetic. Infinity just describes a situation in which our numbers are really huge. A million is nowhere near infinity. A googol is not even any closer. Infinity is just a short way of saying "large without bound." Negative infinity is large in a negative direction, without bound. We cannot really say "small without bound" because that could also describe a limit of zero.
This limit idea is extremely useful. It is one of the basic ideas
of calculus, which is used heavily in most sciences. Limits help us find the
value of pi, and functions like the one on the left. If x is measured in
radians (instead of degrees), then this has a value of 1. By the way, that is
one of the reasons we measure angles in radians, because quite a few
expressions work out so simply with radians.