Why what a function approaches need not be what it equals

0:00A limit is read from what a rule does near a point. Never at it. That sounds like a piece of pedantry, and it is the opposite: it is the reason the whole subject works. Here is the machinery, and there is not much of it. A run of inputs closes in on the place you care about, eight of them, each a tenth as far out as the one before.

0:26Not one of the eight is the place itself. Rounds of elimination then strip away whatever depends on how far out you started, and whatever is left standing is the reading. Five rounds, run on eight values, leave three still to agree with each other. Which matters: when a reading refuses, it refuses because three values disagreed, not because the elimination ran out of things to compare. Start with the case that makes the point hardest.

1:01Take x squared minus four, all over x minus two. At two that expression asks you to divide by nothing, so two is simply not one of its inputs; the number of clauses owning that place is zero. Ask the rule there and it will not answer. Now take the two readings anyway, which you are entitled to do, because neither of them ever asks about two. From below, four.

1:30From above, four. So the limit is four, at a place where the rule has nothing whatsoever to say. No value, and a perfectly definite limit. If that combination were impossible, the rest of calculus would not exist, because every derivative you will ever meet has exactly this shape. The usual move is to factor the top and cancel. x squared minus four is x plus two times x minus two, and the x minus two goes.

2:04What is left is x plus two, which is a perfectly ordinary rule with a value everywhere. Now, is that the same rule you started with? No, and the difference is worth measuring rather than waving at. Sweep eighty-one places across the point. The number of them the quotient will not answer for is one, and that one is two. Of every place they both own, the number where the two forms disagree is zero.

2:34So they differ at exactly one place in the whole sweep, and it is the one place a limit is forbidden to look at. That is why the cancellation is legal here and why it would not be legal in an equation. The tidy form does have a value at two, and it is four, which is the same number the readings gave. Convenient, and as we are about to see, not something you may assume.

3:02Because most of the time nothing dramatic happens at all. Here are five rules, each asked about at one place. x plus ten at five, x cubed at one, three x at two, the rule that hands back three whatever you give it, and x squared plus x at one. Their limits come out fifteen, one, six, three and two. And in every one of the five, the value the rule states at the point is the limit as well.

3:33So substituting works. Twenty-six offered values come with those rules, and the number the formulas fail to reproduce exactly is zero. Two of them are worth pausing on, because they look rounded and are not. Nought point nine nine nine cubed is nought point nine nine seven nought nought two nine nine nine, exactly, and one point nought nought one cubed is one point nought nought three nought nought three nought nought one.

4:04Every digit is real. So here is the question nobody asks. If substitution works, why go through all that? Why not put the point in and be done? Because whether it works is a fact about the rule, and it has to be checked each time. Take all nine of the rules in this video and score substitution honestly. Of the ones that have a limit at all, the number where putting the point in gives you that limit is five.

4:36The number where it hands you a different number, with no warning of any kind, is one. And the number where there is nothing to put in is one. Across all nine, the number of times substitution is not an answer at all is two. It is a method that happens to work, not a definition, and the only way to know which case you are in is to take the readings.

5:03A small thing about those runs of values, and it is worth noticing because it changes how much they prove. Four of them came with the comfortable rules. Look at the distances either side of the point. Of the four, the number with the same count of entries on each side is two. So two of them are simply lopsided: four entries below the point and three above. And both of those carry an extra step at a twentieth, a half-sized step that has no partner on the other side.

5:38Of the four, the number stepping out to the same distances on both sides is one. The odd one has three entries each side and they still do not match: below, a thousandth, a hundredth and a tenth; above, a hundredth, a tenth, and a fifth. None of that is a mistake. But a run of values is evidence, and evidence you did not gather symmetrically is worth reading with your eyes open.

6:09Two of these rules involve a cosine, and that raises a question about the numbers themselves. A cosine is not a fraction, so how can anything here be exact? It can, because cosine has a series whose terms alternate in sign and shrink, and consecutive partial sums of a series like that trap the true value between them. Two fractions, one either side, and no decimal anywhere. At a tenth, the trap runs from nought point nine nine five to two hundred and thirty-eight thousand eight hundred and one over two hundred and forty thousand.

6:50Cut that to four places and you get nought point nine nine five; round it to four places and you get nought point nine nine five. Same answer, so it does not matter which you did. Now go in to a hundredth. The trap is narrower there, a width of one part in two thousand four hundred million. Cut it to four places and you get nought point nine nine nine nine.

7:16Round it to four places and you get one point nought. Those are different answers, and the number you are usually shown is the cut one. The trap fixes three digits comfortably and refuses to fix six, and knowing which digits you have actually earned is not a small thing. Now the case the whole topic is built to reach. Take the rule that is x plus two everywhere, except at one, where it states the value zero.

7:48That is a stated value and not a missing one: the rule is perfectly well defined there, and what it says is zero. Six values come with it, and the number its clauses fail to reproduce is zero, and the number of the six that IS the place being asked about is also zero. Its reading from below is three. Its reading from above is three. So the limit is three, and the value stated at the point is zero, and those two numbers sit three apart.

8:22Both readings exist. Both are perfectly well defined. They are not the same number. Drawn, that gives two marked heights at the point, and the number the rule actually takes there is one. The other mark is where the rule is heading and never arrives. You might suspect the value zero was chosen to be provocative. So change it. Build seventeen rules, alike in every respect except the number each states at that one place.

8:56Seventeen rules, and the count of different values they state there is also seventeen, so no two of them are the same rule. Now take the limits. Across the entire family, the number of different limits is one. Every last one of them has the limit three, whatever it announces in the middle. And of the seventeen, the number whose stated value happens to equal its limit is one. One out of seventeen.

9:26The other sixteen are not broken, or badly behaved, or exceptions. They are the ordinary case, and agreement is the coincidence. Third combination. A rule that hands back one below and at the place, and two above it. It has a value there, and that value is one. Its reading from below is one and its reading from above is two, and asking for the limit gets you a refusal: the two readings differ, so there is no limit.

10:00Nothing here is undefined and nothing misbehaves. Two honest readings simply came back with different numbers. That is the mirror image of the last case: there the two readings agreed with each other and disagreed with the value; here they disagree with each other and the value is beside the point. And the fourth, which is the one people forget to look for. One over x squared, and this time the inputs allowed are only the ones above zero.

10:34That restriction is not decoration. It means that asking for a reading from below is not a question that can be asked: there are no inputs down there to feed it. So the two failures here are different failures. From below, no clause owns that input. From above, the outputs do not settle on any one value. The second one is worth measuring rather than describing. Name any bound you like; eight of them were offered, and the number the run never gets past is zero.

11:08Better than that: the walk in to clear each one takes one step, then two, two, three, three, four, four and five. Every bound falls, and falls quickly. Put the same eight bounds to the tidy rule from earlier and it clears none of them, so this is a test something can fail. No value at the place, and no limit either. Four combinations, then, and every one of them has now happened.

11:38Which is a claim worth being careful about, so here is how it was counted. Nine rules were put to two questions. Does it state a value at the place? Does it have a limit there? Those two questions are answered by separate machinery that never sees each other's answer, and nothing anywhere says which cell a rule belongs in. The cell is just wherever the two answers happen to land it.

12:06Nine rules, four different cells, and the number of the four that nothing reaches is zero. The counts come out one, one, one and six. The crowded cell is the comfortable one, both readings present, and it splits: five where the value and the limit agree, one where they do not. One rule out of nine had no value and a limit; one had a value and no limit; one had neither.

12:35The grid is complete, and not one square of it was filled in by hand. One last thing, about the runs of values themselves. It is tempting to treat a run of six as settling the matter. It does not, and here is a rule that shows why. Build a second rule that agrees with the earlier one at all six of its values: the number it fails to reproduce is zero.

13:03But between a thousandth below the point and a thousandth above, on a stretch a five-hundredth wide, it does something else entirely. The number of the six offered values falling inside that stretch is zero, because the nearest of them sits exactly a thousandth away. Now read it with a run that stops where the values stop, and you get three. Read it with a run that starts four steps further in, and you get nothing.

13:33Same rule, two answers, and the difference is only how close you troubled to get. Of the shallow run's places, the number inside that stretch is zero; of the deeper run's, it is four. The two runs share no places at all. Put the same two runs to the honest rule and both give three. So the ambiguity belongs to the impostor and not to the method. A run of values is evidence, and it rests on an assumption about the gap it never looked at.

14:08So what has all this bought? One rule, stated bluntly: a limit is read near a point and never at it. Everything else follows. It is why a rule with no value at a place can still have a limit there, which is the case every derivative in the rest of this subject will be. It is why substitution is a method to be checked and not a definition. It is why cancelling a factor is allowed, at the one place a limit is not permitted to look.

14:42And it is why the value and the limit are two separate readings that you take separately and compare. Nine rules, four cells, all four occupied. The point is not that functions misbehave. It is that near and at were always two different questions, and only one of them is the one you are asking.