Section 2 Limits 2.0 A Day at the Races: Achilles and the Tortoise

Around 500 BC, there was a Greek philosopher named Zeno. You've probably heard of him, or at least of the dilemma he raised. He stated it in a number of different ways, but I shall concentrate on just one.   The hero, Achilles, and the tortoise agree to a foot race. The tortoise complains that Achilles has the gift of speed, but alas he, the tortoise, does not. So he petitions Achilles for a handicap. And Achilles agrees to allow the tortoise a 1000 meter head start. Surely, Achilles thinks to himself, I can still win this race.

Now Achilles certainly does have the gift of speed and can do a steady 5 meters per second without even breaking a sweat. So that is how fast Achilles decides to run. But the poor tortoise redlines at just half a meter per second. And knowing that he'll have to do his very best even to have a chance against the speedster, Achilles, he must go just as fast has he can go.

The starting signal sounds precisely at noon. The problem is to figure out exactly what time Achilles passes the tortoise.

Well, a little elementary algebra quickly gives us an answer. If xA is where Achilles is, xT is where the tortoise is, and t is seconds past noon, we have the following equations:

æ 1 meters    ö
xT  =  1000 meters +  ç           t ÷    equation of the tortoise
è 2 second    ø

eq. 2.0-1
æ   meters    ö
xA  =     0 meters +  ç 5         t ÷    equation of Achilles
è   second    ø

Since Achilles and the tortoise must be in exactly the same spot when one passes the other, we solve for xA = xT. This being the case, we can subtract the two equations above to get:

æ  1 meters    ö
0  =  1000 meters - ç 4          t ÷                           eq 2.0-2
è  2 second    ø

2
t  =  222   seconds
9

Zeno probably also had the algebraic wherewithall to solve the problem in a similar way. But he observed that after 200 seconds, Achilles runs the 1000 meter distance of the tortoise's head start. At that time he is where the tortoise started. But in that amount of time, the tortoise has run 100 meters. So 20 seconds later (that is at t = 220 seconds) Achilles catches up to that spot. But in those 20 seconds, the tortoise has run another 10 meters. It takes Achilles only 2 seconds to catch up to that spot. But by then, the tortoise has run another meter. Zeno realized that this process goes on forever, with Achilles always catching up to where the tortoise was, and the tortoise always running a little bit farther. So how can Achilles ever pass the tortoise?

We actually have several examples of things that tend to a limit in this parable. We have the sequence of times, t1, t2, t3, etc. At each tj Achilles is where the tortoise was at tj-1. That defines all the tj's except t1, which we will say is the moment of the starting signal. So the various tj's are:

t1 = 0      seconds                                            eq. 2.0-3
t2 = 200    seconds
t3 = 220    seconds
t4 = 222    seconds
t5 = 222.2  seconds
t6 = 222.22 seconds
.
.
.

I'm sure you can see the pattern here, but can you prove it? Yes, you can, and you're going to do so right now.

By what factor does Achilles' speed exceed the tortoise's? Get that by dividing the tortoise's speed into Achille's speed. Call that number r for ratio. Now, if it takes the tortoise an amount of time, Dt, to run some distance, how long does it take Achilles to run that same distance ? Get that by dividing Dt by r. Why? (refer to the rate formula given above).

Don't let the Dt (that is pronounced, "Delta-Tee") throw you. The D just means "difference." So Dt just means the difference between to t's.

Now observe that during each interval, Achilles runs the same distance that the tortoise ran in the previous interval. Here is the pattern of the intervals, which we call delta_t's:

Dt1  =  t2 - t1  = 200 seconds -   0 seconds = 200 seconds
Dt2  =  t3 - t2  = 220 seconds - 200 seconds =  20 seconds
Dt3  =  t4 - t3  = 222 seconds - 220 seconds =   2 seconds
.
.
.                                                              eq. 2.0-4
Notice that the Dt's go down by a factor of 10 each time. Why? Well, you arrived at r = 10 before, didn't you? So it takes Achilles one tenth the time to run the same distance the tortoise did in the previous interval. In that time, the tortoise runs one tenth the distance he did before because he has only one tenth the time to do it in. In the next interval, Achilles will run that distance, and do so in one tenth the time the tortoise did, and so on. And that is why the intervals go down by a factor of 10 each time. So with each interval past, we are effectively adding one decimal place with a 2 in it to the total elapsed time, tj.

We already know that Achilles passes the tortoise at

2
t = 222   seconds                                              eq. 2.0-5
9
from the algebra we did before. Can we show that the sequence of tj's has this as its limit? Well, what do we have to show? We have to show that if I tell you how close you want tj to be to the t we gave above, then you can tell me how far into the sequence of tj's I have to go in order to assure me that all the tj's that follow are at least that close.

You instructor will word it in a different way. He or she will say, "given any e greater than zero, you need to be able to find an n big enough so that  |t - tj| < e   for all  j > n  no matter how close to zero e is."

Don't let the Greek symbol or the mathematical formalism intimidate you. The statement of the problem I gave means precisely the same thing as the one your instructor gives. A limit means simply, I tell you how close I need to be, and you can tell me what I have to do to be that close. In the formal version of this, e (that's Greek letter, epsilon) is the "how close I gotta be" and n is the "what I gotta do to be that close."

If I say, for example, I've got to be within a thousandth of a second of t (i.e.  e = 0.001 seconds), you can tell me, go past t7 (i.e. n = 7) and you will find all values of tj that are within a thousandth of a second of t. If I said I've got to be within a millionth of a second, you could tell me what? (go ahead and extend the table of tj's to find out).

But so far we haven't proved the assertion. To prove it, we must show that we can always find how far to go no matter how close we have to be. Let's do that now.

We can see that the jth term of the sequence given in 2.0-3 is a bunch of 2's, beginning in the hundreds column and extending rightward. There are always j-1 2's in each decimal. So we can easily write the decimal for any term in the series. The 13th, for example, is: t13 = 222.222222222 (count 'em -- twelve 2's).

It's pretty clear that after t3 all the tj's are within 1 of 222. So all we will prove is that the fractional part of the sequence (that is the part to the right of the decimal point) converges to a limit of 2/9.

So let's find the difference between a bunch of 2's to the right of the decimal point and 2/9. Recall that if we have n 2's to the right of the decimal point, what we really have is the counting number made of n digits of 2's divided by 10n. So, for example, nine 2's to the right of the decimal point means:

222222222
0.222222222  =                                                 eq. 2.0-6
109
So, in the example where n=9, we are interested in:
ç 2     222222222 ÷
e  = ç    -            ÷                                      eq. 2.0-7
ç 9        109    ÷
e represents how close we are to the limit.

We have to put these fractions over a common denominator in order to subract them. So to the right-hand one we have to multiply top and bottom by 9. To the left-hand one we have to multiply top and bottom by 109. We get:

ç 2000000000     1999999998 ÷          2
e   =  ç             -             ÷  =                        eq. 2.0-8
ç 9000000000     9000000000 ÷      9000000000
In fact, when you multiplied those nine 2's by 9, it became pretty clear that there is a pattern there. And if you multiply n 2's by 9, what you will always get is a 1 followed by n-1 9's, followed by an 8. And when you subtract that from 2 × 10n, you will always get 2. So that will always be the numerator of the difference. And the denominator? Clearly it will always be 9 × 10n. So for a decimal point followed by n 2's, we have:
2
e  =                                                            eq. 2.0-9
9×10n
So, can you make e as close to zero as you like by choosing the right n? and does e stay at least that close for all subsequent n? The answer is clearly yes, because the numerator doesn't change as n gets big, but the denominator keeps getting more and more huge.

Well we have diverged a long way from the foot race. So what have we proved? We have proved that the sequence of times, tj, does indeed converge to a limit, and that limit is the same time that we predicted earlier, using algebra, that Achilles should be passing the tortoise. We can tell Mr. Zeno that even if he can never see how Achilles gets to 222 2/9 seconds, we can show that he certainly gets as close as anybody could possibly ask.

And since we know that Achilles does indeed pass the tortoise, our notion of a limit bears a true connection to reality. Zeno's dilemma was that the sequence of times leading up to the passing represented a completed infinite, yet the infinite is defined as that which never ends. Zeno's problem was that he had a poor definition of "infinite." Infinites do end all the time. And they end in a limit.