Section 2 Limits

2.1 How Close to the Edge Dare You Go?

Hemera ©
When I was a kid, my aunt went and bought this tiny little car. Undoubtedly the real reason she did it was because she was going through some financial difficulties at the time, and it was all she could afford. But what she told her kids (my cousins) was that small cars were good. Big cars hog the road, don't drive well, destroy the earth, and so on. Well one day my aunt and cousins showed up packed into this thing. The eldest of my cousins got out and proceeded to repeat the whole "small car good, big car bad," line to me. We then walked down the driveway, and there was our big old station wagon, which my dad had assured me was a great car. So I asked my cousin, "Is this car too big?" Realizing that a little tact was called for, he replied, "No, but if it were any bigger, it would be too big."

Well, I found that response perfectly maddening. You mean, if you added just one inch, it would be too big? What if you just added a coat of paint to the bumper -- that would make it a little bigger. Would it be too big then? It was clear to me that there was no place where too-bigness started, only a place where small-enough ended. What was maddening had nothing to do with my cousin. It had everything to do with discovering something new about how the world works. What was the biggest a car could be and not be too big? Our station wagon filled that bill. But what was the smallest a car could be and still be too big? In my cousin's world, no such thing could possibly exist.

It took days for me to resolve this dilemma in my own head.

Hemera ©
And what resolved it was thinking about all the little-boy macho games I had played with other boys -- like the time a bunch of us found our way to the roof of a condemned building and dared each other how close to the edge of the roof we could stand. Clearly there was a place where you could not stand, and if you tried to stand there you would find yourself plummeting toward the ground. But there was no place that was the absolute limit of where you could stand without falling, and that, of course, was the whole point of the game. No matter how close you stood to the edge, some other guy could stoke up his courage and skooch just a tad closer to disaster than you were.

And this brings us to another kind of a limit. How close can you stand to mathematical disaster without falling off the edge? But what is mathematical disaster? Well, division by zero, for one. Thou shalt divide by any of the fruits of this garden, but by the tree of zero, thou shalt not divide. You mean I can divide by a tiny little number like 0.00000001, but not by zero? Yep. Even by 0.0000000000001? Yep -- even tinier than that. Just don't go dividing by zero.

So there is that spot we can't stand on without falling. But if I dare you to stand closer to it than I am, you can always do it.

So suppose I had the function:

          x²  - 3x + 2
   y  =                                                           eq. 2.1-1
             x - 1

Well, you can probably quickly convince yourself that the numerator of 2.1-1 factors and you have the same as:

         (x - 1) (x - 2)
   y  =                                                           eq. 2.1-2
              x - 1

And of course the x - 1 on the top cancels with the same thing on the bottom right? and you're left with:

   y = x - 2                                                      eq. 2.1-3

Well, kind of. That works for all values of x except x = 1. When x = 1, the denominators of 2.1-1 and 2.1-2 become zero, and so you have no function at all. In other words, the domain of this function does not contain the point, x = 1. So if we graph this function, we have a straight line with slope 1 and intercept -2, but with a hole in it at x = 1 (see figure 2.1-1). Fig. 2-1.1: Plot of (x - 1) × (x - 2) / (x - 1)

Fig. 2-1.1: Plot of (x - 1) × (x - 2) / (x - 1)

The figure shows the missing point at x = 1. But observe that you have a notion of what that point ought to be. If you just reached in with your felt-tip pen and blotted at (1, -1) you have a feeling that you would be completing the graph.

Let's go back to the macho boys standing on the roof of the condemned building, but imagine that it's a sloped roof. So the altitude of each boy's feet varies according to where on the roof he is standing. Based upon how far right or left a boy is standing, we can determine how high his feet are. Imagine that red trace of our graph represents the roof. The hole in the graph is a hole in the roof. If one of the boys tries to stand there, he will fall all the way to the basement and his mother will be angry with him. But the boys persist in a game of who-can-stand-nearest-the-hole. So how high would a boy's feet be if he could stand on the hole?

In a way, that sounds like an absurd question -- rather like the proverbial tree falling in the woods with nobody around. You could even be a smartass about it and say his feet would be at the same altitude as the rest of him, all twisted up in the basement. Yet we have a sense that there is a serious answer to this based upon the behavior of the rest of the roof.

Let's say we measure altitude from the basement, which on our graph we'll say is at y = -10 meters (that is the basement is 10 meters below the x-axis), and that each square on the graph is 1 meter. We sense that the answer to how high a boy's feet would be if he could stand on the hole ought to be 9 meters. But why?

We can learn much from the game they are playing. We see that the closer a boy stands to the hole, the closer his feet are to being exactly 9 meters above the basement. How close to 9 meters above the basement would you like a boy's feet to be? No matter how close that is, you can dare him to stand close enough to the hole that his feet will be within that degree of closeness to 9 meters above the basement.

And that's the point. If you can tell me how close the boy's feet have to be to 9 meters, I can tell you how close he has to stand to the hole. That is what makes 9 meters a limit. So we can say, the limit of altitude as the boy's position goes toward the position of the hole is 9 meters. Or to express it in the standard math shorthand:

                  lim                  altitude of his feet  =  9 meters
    boy's position → position of hole

Or to put it more traditionally, let x be the boy's position. Let x_h be the position of the hole. Let a_f be the altitude above the basement of his feet. Then we write:

      lim   a_f  =  9 meters
     x → x_h

Now let's analyze equations 2.1-1 and 2.1-2 again. As we already observed, everywhere except at x = 1 either of these functions is identical to y = x - 2. At precisely x = 1, the functions given in equations 2.1-1 and 2.1-2 have no meaning. But as x gets very close to 1, y = x - 2 gets very close to -1. Not only that, you can get y = x - 2 as close to -1 as you'd like simply by skooching x close enough to 1. We can see this trend without ever having to evaluate y at the forbidden value of x = 1. And so we say, the limit of y as x goes toward 1 is -1. Or using the mathematical shorthand:

           (x - 1) (x - 2)
     lim                     =  (x - 2)  =  -1
    x → 1      (x - 1)

It is with near certainty that your teacher will call upon you to prove this or that limit is equal to such and such. And he or she will want you to use the traditional method of delta and epsilon (that is the method of δ and ε) to do it. Before I demonstrate that for you, I want you to make the connection between this and what we have been talking about. In the case of the boy on the roof, I could tell you how close I wanted the altitude of his feet to be to 9 meters, and you could respond with how close to the hole he had to be. That made 9 meters the limit. In the case of equations 2.1-1 and 2.1-2, I could tell you how close I wanted y to be to -1 and you could tell me how close x had to be to 1 in order to make that happen. That made -1 the limit for y.

In the δ and ε proofs, the ε is the how close the dependent value has be, and the δ is the how close you have to make the independent variable to the "goes toward" value in order to make that happen.

In the case of the boy on the roof, the ε is how close his feet are to being exactly 9 meters above the basement, and the δ is how close to the hole he has to stand in order for that to happen.

In equations 2.1-1 and 2.1-2, the ε is how close y is to -1 and the δ is how close x is to 1.

The typical way you see it phrased in a math text is:

We say that:
              lim    f(x)   =   y
             x → a
if and only if for each ε > 0 there exists a δ such that |f(x) - y| < ε whenever 0 < |x - a| < δ .

(Note that |x - a| is greater than zero and therefore never is zero. We are not interested in the behavior of f(x) where x is equal to a. What we are interested in is the behavior of f(x) when x is very close to a)

Don't let the formalism intimidate you. What it means is this: You have some real valued function of x, and for convenience we'll call it f(x) (that is it returns some real value for each real value, x, in its domain, that you feed it). f(x) could be almost any expression of x you can think of. You want to see if f(x) approaches some value, y, as x approaches some other value, a.

The test is, I name how close to y I want f(x) to be. That is I say I want f(x) to be within ε of y. If, no matter how small an ε I name, you can name how close x has to be to a (without actually being equal to a) in order to guarantee what I asked for, then you have proved the limit.

So in the example of equations 2.1-1 and 2.1-2, we have:

             (x - 1) (x - 2)
    f(x)  =                                                       eq. 2.1-3
                 (x - 1)

We also have a = 1 and y = -1. If I say that f(x) has to be within 0.1 of y, you can say, "make x within 0.1 of a". If I say f(x) has to be within 0.0000001 of y, you can say, "make x within 0.0000001 of a." In general, if I say that f(x) has to be within ε of y, you can say, "make x within δ of a," where, in this example, δ = ε (don't get the idea that they are always equal. In most of the proofs of this kind that you will be assigned, there will be a relationship between them that you will have to find, but it will seldom be simple equality).

In general, if you can show a formula that, given an ε, turns out a δ that always meets the test, then you have met your teacher's test, and he or she will mark your work correct.

The Main Point to Remember from 2.0 and 2.1

We have discussed two kinds of limits so far, a limit that a sequence of numbers approaches, and a limit that a function approaches. They may seem very different, but they aren't. In both cases, a limit is like a contract. It says, if you need to be this close to something (and it doesn't matter how close this close is), then I can tell you what you need to do to be this close. That is what the delta-epsilon proofs are all about. That is what limits are always about.

The first type of limit is similar to a race-car mechanic telling the machinist how accurately the engine parts must be cut, and the machinist telling him how many dollars it will cost. However accurate the mechanic wants it, the machinist can deliver it with that accuracy. But the tighter the tolerance, the more the cost.

The second is similar to aiming a rifle. You tell the marksman how close to the bullseye he has to hit, and he can tell you how still he has to hold the rifle in order to make that happen.

Here is a graphic that illustrates the delta-epsilon method of proving the second kind of limit. The two green horizontal lines are no more than ε above the limit as x goes toward 2 and no more than ε below the limit. The two blue vertical lines are spaced δ on either side of x = 2. A delta-epsilon proof requires that you show that you can always make the two horizontal lines as close as you like to the limit provided you make the two vertical lines sufficiently close to whatever x is going toward (in this case 2). As you bring the two vertical lines closer and closer to what x is going toward, the two horizontal lines must close in on the limit.

If you remember these underlying ideas about limits, learning the details of how to do limits proofs will be a lot easier for you.

Worked Example and Exercises

Section 2.2: More on Limits

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