Approaching from the left and from the right, and when the two disagree

0:00Pick a point on the number line and start walking towards it. There is no single way to do that. You can come up from below, or you can come down from above, and those are two different journeys that happen to end at the same place. So a rule fed inputs on the way in has two stories to tell about that point, not one. Most of the time the two stories agree and nobody notices there were ever two of them.

0:29This is about what happens when they do not. And the surprise is not that the two answers can differ. The surprise is what that costs: once they can differ, the single answer you were hoping for stops being something a rule simply has, and becomes a verdict that has to be earned. So here is how a reading is taken, and the whole topic sits in one detail. Feed the rule a run of inputs closing in on the point from one side: a tenth away, a hundredth, a thousandth, and on down.

1:07Look at what comes out, and take whatever those outputs are settling on. Not one of those inputs is the point itself. The point is never fed in, not once, so whatever the rule does when it gets there cannot possibly reach the answer. That is not a convenience; it is the definition, and everything strange in this topic follows from it. Do the same thing from the other side and you get a second reading, computed without any reference at all to the first.

1:41Two runs, two answers, taken independently. Nothing so far has told you they must match. Take the plainest rule that can misbehave. Below zero and at zero it hands back one; above zero it hands back two. That is a complete definition, with nothing vague about it. Ask what it says exactly at zero and it says one, because exactly one of the two clauses owns that place, and it is the first.

2:14Now take the two readings. Coming up from below, every input meets the first clause, so every output is one, and the reading is one. Coming down from above, every input meets the second clause, so every output is two, and the reading is two. Ask this rule for its limit at zero and it does not hand back a number at all; it says the two readings differ, so there is no limit.

2:42And it refuses for that reason specifically, not because either reading failed to settle: both settled perfectly well. Now let us prove the thing that was claimed a moment ago rather than just repeating it. Take the same two clauses and move the inclusive sign onto the other one, so the boundary now belongs to the second clause instead of the first. Ask at zero and the answer changes: it was one, and it is now two.

3:16Take the two readings again. One from below and two from above, exactly as before. The value at the point moved and the readings did not budge, which is what it means to say the readings never consult it. And here is the sharpest version of the same point. Build a rule out of two clauses that between them leave the boundary out altogether, so that the number of clauses owning that place is zero.

3:45Ask it there and it will not answer: no clause owns that input. Ask for the two readings and they are two and two, and the limit is two. A rule that says nothing whatsoever at a point can still have a perfectly definite limit there. Draw the jumping rule and the whole argument becomes a picture. A flat run at height one coming in from the left, a flat run at height two going out to the right, and a break at the vertical line where they meet.

4:18At the break there are two marks, and they are not decoration. One is filled in, sitting at height one, and it records the value the rule actually takes there. The other is hollow, sitting at height two, and it records a reading: a height the outputs head towards from the right and never once reach. Of those two heights, the number the rule genuinely takes at that place is one.

4:46That is what a hollow circle has always meant. It is not saying the rule is undefined; it is saying this height is approached and not attained. A filled mark is a value, a hollow mark is a reading, and confusing the two is the single commonest way to misread one of these pictures. Now the rule for the verdict, and notice it is a consequence and not a definition handed down.

5:16Take the reading from below. Take the reading from above. Compare them. If they are the same number, that number is the limit. If they are not, there is no limit, and that is not a failure of technique or a sign that you should try harder. Both halves succeeded; they simply disagreed. Which means the phrase the limit exists is never something you assume before you start. It is the last thing you find out, after two independent computations have been laid side by side.

5:52Everything else in this topic is an application of exactly that. Here is a second rule of the same kind, arranged to make one particular point unavoidable. Below zero it is x minus two; above zero it is x plus two; and at zero itself it is nothing at all. Six values are offered to look at, three on each side, and not one of the six is the point being asked about.

6:23Minus a hundredth gives minus two point zero one; a thousandth gives two point zero zero one. Of those six, the number the clauses fail to reproduce exactly is zero, so the offered values really are this rule and not a sketch of it. The reading from below is minus two, the reading from above is two, and the limit does not exist. But the rule does state a value there, and that value is zero.

6:54It sits two units from one reading and two from the other, stranded in the gap between them. Drawn, that gives three marked heights at the break, and the number the rule actually takes is one. Defined here, and no limit here, in the same picture. Now the trap that a table of values sets, and it is worth measuring rather than warning about. Build a second rule that agrees with that one at every offered place below zero.

7:25The number of those three places where the two differ is zero: below the break they are indistinguishable. Their readings from below are the same too, minus two and minus two. So everything a table of values below the point could ever show you is identical for both. And one of them has a limit at zero, of minus two, while the other refuses. The difference lives entirely on the other side, where the table never looked.

7:57Which is why tabulating on one side settles one reading, and one reading is half a verdict. Not weak evidence for the whole thing: half of it, and the half you skipped is where the disagreement was hiding. Two more rules, and they are the ones students expect to behave differently. The size of x divided by x, and x divided by the size of x, each given the value zero at the origin.

8:28Sweep a hundred and twenty places either side of zero and ask whether the two ever disagree: the number of places where they do is zero. Away from the origin they are one function wearing two spellings. Across all those places the first of them takes exactly two different values, minus one and one, and nothing in between. So its reading from below is minus one and its reading from above is one, and the same for the other.

9:01Of the two rules, the number with a limit at the origin is zero. And both of them do state a value there, and it is zero for both, which changes nothing whatsoever. Turning the fraction upside down was never going to help, because the two sides were disagreeing about a sign, and a sign survives being inverted. Here is the mirror-image mistake: assuming that two different formulas must produce a disagreement.

9:33Take a rule that is two x plus three below zero and three times x plus one above it. Those are genuinely different expressions. Across the swept places, the number where the two formulas would give the same answer is zero, so they really do not coincide anywhere. And yet the reading from below is three, the reading from above is three, and the limit is three. Different roads, same destination.

10:03Ask the same rule at a place well inside its second clause and the limit is six, with only one formula in play. Now a rule shaped the same way: x squared minus one up to one, and minus x squared minus one after it. Its readings at one are zero and minus two, and there is no limit. Of those two rules, the number with a limit at its own boundary is one.

10:31Same shape of question, opposite answers, and the test was never on the expressions. It is on the two numbers they head for. So far every question has been asked at one place. Ask it everywhere instead. Take the rule that is the size of x plus one below zero, minus one less than the size of x above it, and zero at the origin, and put the verdict to it at forty-one stops along the line.

11:04The number of stops where it refuses is one, and that one is the origin, where the readings are one and minus one. Everywhere else the limit exists and, away from that one bad stop, the number of places where the verdict is not the value the rule itself states is zero. Does the verdict always agree with the stated value where both exist? It does not, and here is a rule that shows it.

11:33It is simply x, except at one, where it announces five. Its verdict at one is one, and the value it states there is five. Across those same forty-one stops, the number where the verdict and the stated value part company is one, and that is where. Two computations, agreeing almost everywhere, and never obliged to. Let us make the central point as blunt as it can be made. Build nineteen rules that differ from each other in one respect only: the number each announces at the boundary.

12:11Nineteen rules, and the count of different values they announce there is also nineteen, so every one of them is a genuinely different rule. Now take the readings. Across the whole family the number of distinct readings from below is one, and from above it is one. Every last one of them reads minus two from below and two from above, whatever it announces in the middle. Of the nineteen, the number with a limit at that boundary is zero.

12:43Two of them are cunning: their announced value is exactly equal to one of the two readings, and it still buys them nothing, because the other reading has not changed either. Being defined at a point is a fact about the point. A limit is a fact about the places around it, and the two are answering different questions. So take stock of what those readings actually decide. Six rules were put to them here.

13:14The number where a reading failed to settle on anything at all is zero: every reading in this video succeeded. The number where the two readings then disagreed is five. And the number of those six that do state a value at the very place in question is six. All six defined, five with no limit. That triple is the whole topic in three numbers. Nothing went wrong, nothing was undefined, nothing misbehaved.

13:45Two honest computations were carried out and they came back with different answers, and a limit is precisely the thing you are entitled to write down when they do not. So there are two roads into every point on the line. Walk both. The one you skip is the one that was going to disagree with you.