Substitution works until it gives nothing over nothing, and then cancellation does

0:00There is a move you are about to be taught, and it is so easy that it hides what it costs. To find what a rule settles on as the input closes in on a place, put the place in and read off the answer. For the rules in front of you that is usually right, and it is not a definition; it is something that has to be earned.

0:23So nothing here will ever put the place in to find out what a rule settles on. Every reading is taken from twenty-eight inputs closing on the place from one side, none of them the place. A model is fitted to at most twenty-six of them, and then made to predict the two it was never shown. A model that cannot predict them is thrown away, and a model that runs off to nothing underneath at the place is refused.

0:53That way, when putting the number in turns out to agree, the agreement is a finding rather than an assumption. And when it disagrees, we will see that too. Start with the smallest possible claim: as the input closes on a place, x itself settles on that place. Now the step, and it is one line. If x to the k settles on the place to the k, then x to the k plus one is x to the k times x, and the product rule multiplies the two answers together.

1:30That is an induction, and it is worth running rather than quoting. Twelve powers, at a place of three halves, each read on its own from its own run, and not one of them ever evaluated at the place. Then eleven steps: for each, compare the reading of the next power against the reading before it multiplied by the reading of x. The number of the eleven that fail is zero, and the last of them lands on five hundred and thirty-one thousand four hundred and forty-one over four thousand and ninety-six.

2:08That zero is only worth something if it could have been otherwise, so run the same eleven steps again with the wrong thing multiplied in. Now all eleven fail, and the last lands on eight hundred and eighty-five thousand seven hundred and thirty-five over the same four thousand and ninety-six. And to be sure the readings really are readings, a thirteenth rule is thrown in that is the twelfth power away from the place and one more than it close in: it reads exactly one more, which a power put in by hand never would.

2:47The agreement was earned. A polynomial is a sum of constant multiples of powers, so three rules carry it: the sum rule splits it, the constant rule pulls each number out, and the power result finishes each piece. Take eleven terms, all of them plus one times a power, and drive the input toward minus one. Read the whole thing at once and it settles on one. Now take it apart: the eleven pieces read one, minus one, one, minus one, and so on down the line.

3:25They are not all the same number - there are exactly two distinct values among them - so adding them up is doing real work. Add them and you get one, which is what the whole thing gave. So a polynomial settles on the number you would get by putting the place in, and now you know why: three rules, applied in order, each one licensed before it was used.

3:51That is the theorem, and this is where substitution becomes legal. Now a quotient of two polynomials. The quotient rule needs one thing: the bottom must not settle on nothing. Take x squared plus one over x plus a hundred, and drive x toward one. The bottom settles on a hundred and one, which is comfortably not nothing, so the rule applies and the answer is two over a hundred and one.

4:22Putting the number in gives the same thing, and that is not a coincidence - it is the polynomial theorem used twice and then divided. This is the easy branch, and it is worth naming as a branch rather than as the whole story. Everything interesting happens when the bottom does settle on nothing. And there, two quite different things can be going on. First case: the bottom dies and the top does not.

4:52Take one over x minus two, driving x toward two. The top is stubbornly one; the bottom goes to nothing. The outputs do not settle on anything, and the honest record of that is that there is no answer - not that the answer is enormous. That distinction is worth measuring rather than asserting. Offer the run six bounds, ten, a hundred, a thousand and on up, and ask how many of them the outputs never get past.

5:25The number is zero: it clears all six, one step in, then two, then three, four, five and six. Put the same six bounds to a rule that does settle and it clears none of them. So running past every bound is something a run can fail to do, and this one does not fail. There is no number here, and saying the answer is infinite would be putting a name where there is nothing to name.

5:54Second case, and this is the one worth slowing down for. The bottom dies, and so does the top. Put the number in and you get nothing over nothing. That is not the answer, and it is not a failure either. It is a diagnosis: it says the place is a root of the top and a root of the bottom, so the same factor is sitting on both sides of the line.

6:20How many times it sits there is a question with an answer, and the answer decides everything that follows. Count the factor out of the top, count it out of the bottom, and compare the two counts. That comparison is the whole of what comes next, and nothing over nothing was the thing that told you to make it. Here is the move that makes the diagnosis useful, and the objection that always comes with it.

6:50You cancel the shared factor - but that factor is nothing at the place, and you were taught never to cancel nothing. The answer is that at the one place where it is nothing, the limit is not looking. So score it rather than argue it. Take one such quotient and its cancelled form, and compare them at eighty-one places spread either side. The number of places where both have a value and the values differ is zero.

7:20The number of places the original will not answer for is two; for the cancelled form it is one. So there is exactly one place where one of them has a value and the other does not - and that place is two, the very place the question is about. The cancelled form says nothing there. Two forms, identical everywhere the limit looks, differing only where it is forbidden to look.

7:48That is why the cancellation is ordinary arithmetic and not a sleight of hand. Now watch how little nothing over nothing tells you on its own. Four quotients, all of them nothing over nothing at the place. The first, whose top carries the shared factor twice and whose bottom carries it once, settles on nothing. The second is those same two the other way up, and it settles on nothing at all - there is no answer.

8:19The third settles on minus four. The fourth, after two fractions are combined into one, settles on two. Nothing, no answer, a negative number, a positive number - from four starts that look identical. If the form told you anything, those four would not be able to end so far apart. It is a signal to factorise and nothing more. That fourth one deserves its own look, because the first move is not a limit at all.

8:52Two fractions are subtracted, so put them over one bottom first and only then ask the question. Combine them and the top comes out as x squared minus four x plus three. It is worth doing that step slowly, because a single slip in it changes everything downstream. Write x minus two where the working needs it and squaring gives x squared minus four x plus four, and taking one away leaves x squared minus four x plus three.

9:26Write x squared minus two by mistake and squaring gives x to the fourth minus four x squared plus four, and taking one away leaves x to the fourth minus four x squared plus three. Those two differ by x to the fourth minus five x squared plus four x, which is not nothing, so only one of them can be right. The one that is right is the one whose top, before any factor is taken out, is exactly that combined line multiplied by what it sits over - and the difference there is nothing at all.

10:04Take the shared factor out and it comes out twice from the top and twice from the bottom, leaving x times x minus three over x squared times x minus two. Which settles on two. So the comparison of the two counts has three outcomes, not two. More of the factor on top, and the extra copies survive the cancelling and make the answer nothing. More underneath, and copies are left downstairs with nothing to kill them, so there is no answer.

10:39The same number both sides, and everything cancels, leaving a quotient of two numbers that is finite and generally not nothing. The way this is usually set out covers the first two and passes over the third in silence. That is worth counting. Across the fourteen quotients scored here, two have the higher count on top and three have it underneath. Nine have the same count on both sides. So the two cases that get stated between them settle five of the fourteen, and leave out nine.

11:15The case usually left out is the one most of these problems are in, and it is the only one of the three that can hand you a number that is neither nothing nor missing. Everything the top-heavy ones settle on is nothing, and not one of the bottom-heavy ones settles on anything at all. Time to score the move we started with. Sixteen rules, each asked twice: once by a run closing in, and once by putting the number in.

11:48Neither side is ever shown the other's answer. Five times the two agree. Five times only the run answers, because putting the number in hits nothing underneath and stops. Once only the putting-in answers, and the run is the one that stops - a case we will come back to. Three times neither answers at all. And twice the two disagree. Those two are rules that behave like x plus two everywhere except at one place, where they announce something else: five in one case, nothing in the other.

12:25The run reads three both times, because three is what the rule is doing nearby. Putting the number in reads the announcement, which is a fact about one place and not about the approach. That is the whole argument in one line: substitution reports what a rule says AT a place, and a limit is about what it does NEAR one. There is one more shape worth having in your hands.

12:53The difference of two like powers, divided by the difference of the two numbers themselves. Divide it out and there is no remainder: across eight powers, the number of divisions leaving anything over is zero. What comes out has exactly as many terms as the power: one term, then two, then three, on up to eight. In each term the power on the variable and the power on the place add up to one less than where you started, and the number of terms across all eight that break that pattern is zero.

13:31So at the place every term is the same, and there are as many of them as the power. At a place of two the eight come out one, four, twelve, thirty-two, eighty, a hundred and ninety-two, four hundred and forty-eight and a thousand and twenty-four. Read the same eight from their runs, dividing nothing, and you get the same eight numbers. Turn it on a quotient of two such shapes and the two divisions leave fifteen and ten, so the answer is three halves - which the run confirms.

14:06One last case, and it is not a polynomial at all. A square root minus one, all over the amount you moved. Nothing here is allowed to be a decimal, so the run is chosen to keep the root exact. Step by an amount whose square makes the inside a perfect square, and the root is a fraction every time. Fifty-six places, on both sides. The number of them that ARE the place is zero, the number where the root fails to be exact is zero, and the number that land on the wrong side of the place is zero.

14:43The widest step is twenty-one hundredths and the narrowest is minus nineteen hundredths, and the run closes in from there. It settles on one half. That agrees with what the power shape gives when the power is a half, and that extension is worth flagging: it is stated without proof and used on trust. One of the few places in this material where you are asked to take something as given rather than watch it earned.

15:15So the recipe is short. Put the number in; if the bottom survives, you are done. If the bottom dies and the top does not, there is no answer. If both die, count the shared factor out of each, compare the counts, cancel, and read what is left. Now the honest part, which is about the checking rather than the mathematics. Every reading here came from fitting a model to a run and making it predict places it had not seen.

15:46That works for ratios of polynomials, which is everything in this material, and it fails outside that family. A rule whose values depend on which step of the run you are on is refused, and the refusal says the fit found nothing - which is a different sentence from the outputs not settling. Both sentences get said here, and they mean different things. There is a rule that behaves like x plus two except very close in, where it does something else: six of the shallow run's places sit outside that stretch and none of the deep run's do.

16:26A model fitted to the outermost places alone would have announced three; made to predict the rest, it reads four, which is what the rule actually does. That is the difference between evidence and proof, and it is the same difference the whole of this rests on.