Limit — where it appears
Named by 37 essays across 4 fields — each of them below, with the objects they name alongside it.
Adding up rectangles until they stop being rectangles
The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.
The curve that is its own slope
There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.
A bell curve assembled out of coin flips
Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.
A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
A sum whose terms vanish and whose total does not
Add a half, a third, a quarter, and keep going. The terms shrink to nothing and the total passes every number there is — but so slowly that no computation will ever watch it happen.
The slope of a single point
A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.
The same terms, in a different order, adding to whatever is asked
Flip alternate signs in the harmonic series and it converges. Reorder the terms — add nothing, remove nothing — and it converges to any number chosen in advance. Addition stops being commutative, and the picture shows where it goes.
Area is the undoing of slope
Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
The rule that forgets where it came from
A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.
The sum that fits in one square
Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.
The staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
When the period grows without bound
A repeating signal has a spectrum of separate lines. Stretch the gap between repeats and the lines crowd together while the curve they sit on stays exactly where it is — and at infinite period the lines are gone and the curve is the whole answer.
The walk that becomes a curve
Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.
A map that shrinks everything
One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.
A rectangle grown on two sides
A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.
The flat map that fits closest
A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.
The slope of the mirror image
Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.
The equation with only one answer
A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.
The constant that counts what does not happen
Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.
A curve that has area
The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.
A ball whose outside is not one
Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.
No interval in it, and length to spare
The middle-thirds set has no length because the removed pieces add to one. Remove shrinking middles instead and they add to a half — leaving a set that still contains no interval anywhere, and still has half the length it started with.
Covering a set from outside
To say how long a set is, cover it with intervals and add their lengths, then take the smallest total any covering achieves. That definition is short, obviously right for an interval, and gives the rationals a length of nothing.
Which functions can be added up
Riemann's integral works when the upper and lower sums close on each other. The exact condition for that, found once measure existed to state it in, is that the points where the function jumps have measure zero — which some nowhere dense sets fail.
The time spent and the share held
Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.
Where the shares have nowhere to go
On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.
How many get their own hat
The chance that nobody gets their own hat settles on 1/e. The chance that exactly one person does settles on 1/e too, exactly two on 1/(2e), exactly three on 1/(6e) — the Poisson distribution with mean 1. The reason is a set of averages that come out exactly 1 at every size, and the counts reach the limit so fast that eight hats are within six ten-thousandths of it.
Which curves have a length at all
A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.
The length belongs to the journey
Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.
The series everything else is measured against
A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.
The repair at the boundary
Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.
Pinned between two sequences
The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.
The slice that has to match
The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.
An endless region with a finite area
A region that runs off to infinity can still have a finite area, and for the curves 1/xᵖ the exponent that makes the far end finite is exactly the one that makes the end at zero infinite. 1/x fails at both, no power succeeds at both, and a horn can hold less than π while needing infinite paint.
A round table with no couple together
Seat n couples round a table, men and women alternating, so that nobody sits beside their partner. Once the women are placed the men face a board of forbidden cells that bends round a corner — and that corner is the whole difficulty. The forbidden cells form a cycle, a count of non-adjacent points on a cycle finishes the problem, and the chance of a good seating creeps towards e^(−2) far more slowly than the hat problem reaches 1/e.
The length the derivative never sees
The Cantor function climbs from 0 to 1 with a slope of zero almost everywhere, so the formula ∫√(1 + f′²) dx says its graph has length 1 — the length of a flat line. The inscribed polygons say 2. Mix it half and half with the diagonal and the length becomes exactly the golden ratio, while the formula still reports only the part the slope can see.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvergenceDerivativeContinuityApproximationGeometric seriesHarmonic seriesAreaCantor setConvergence rateCounterexamplee, the numberLogarithm