The Limit

The slope of a straight line is one number and it is the same everywhere along the line. A curve has no such number. It is steeper in some places than others, which is the whole reason it is a curve. So start with the question that does have an answer.

Take two points on the curve, one at input a and one at input b. Draw the straight line through them. That line cuts the curve at both points, which makes it a secant line A straight line that cuts a curve, meeting it at two or more points. For a circle it is exactly two, because a straight line cannot cross a circle three times. The extended radius of the unit circle is a secant line, and the secant function is the distance measured along it from the origin to the tangent line., and it has an honest slope: the height gained divided by the distance covered, or (f(b) − f(a))/(b − a).

Both of those differences get a name and a symbol. Δ is the capital Greek letter delta and it is read as "change in". Δx A change in the input of a function, written Δx, with Δ the capital Greek letter delta read as "change in". It is an ordinary number with a size: set it to 0.1 and it is the width of a bracket you can draw on the picture. Δy is the matching change in the output, and Δy/Δx is the slope of the straight line through the two points, which is subtraction and division and needs no limit at all. Setting Δx to zero puts both points in the same place, so Δy is zero too and the quotient becomes 0/0. is the change in the input, b − a. Δy is the change in the output, f(b) − f(a). The slope is Δy/Δx, and both of the letters in it stand for a gap you could measure with a ruler.

The curve here is f(x) = x², and the next step stays on it.

None of this is calculus. (f(b) − f(a))/(b − a) is a subtraction, a subtraction and a division, and every one of those is defined. It is the average rate of change over the stretch from a to b. Nothing here needs a limit and nothing here should be read as hinting at one.

Notice what it does not give you. The answer depends on both points. Move either one and the tilt changes, so the number belongs to the pair, never to a single place on the curve. There is no answer yet to "how steep is this curve at x = 1", only to "how steep is this curve between x = 1 and over there". The next step tries to close that distance.

Drive the second point in toward the first before reading any further. The control walks it down a ladder of gaps, and the last rung of that ladder is zero.

Two things happen at once and both matter. The first is the line. With the fixed point at x = 1 on f(x) = x², a gap of Δx = 1 gives a slope of 1. At Δx = 0.5 the slope is 1.5. At 0.1 it is 1.9, at 0.01 it is 1.99, at 0.001 it is 1.999. Somewhere down that ladder the line stops moving in any way the eye can detect, and it is plainly sitting at a definite tilt.

The second is the fraction. Δy is shrinking too, and faster. At Δx = 0.1 the change in height is 0.19. At 0.01 it is 0.0199. At 0.001 it is 0.001999. The top and the bottom of the fraction are both marching down to zero while the quotient between them sits calmly near 2.

Now take the ladder to its last rung. At Δx = 0 the two points are the same point. Δy is 0, Δx is 0, and the panel refuses to print a slope. That is not the figure failing. It is the formula having nothing to say.

Here is why 0/0 has no value, and it is not the same reason 1/0 has none. A quotient a/b = c is a claim that c · b = a. For 1/0 there is no c at all, because anything multiplied by 0 is 0 and never 1. For 0/0 the problem is the opposite: every c works, since 0 · 5 and 0 · 17 are both 0, and nothing in the expression chooses between them. An expression that fails this way is called an indeterminate form An expression such as 0/0 whose parts do not decide its value. A quotient a/b = c is the claim that c · b = a, so 1/0 has no value because no c satisfies c · 0 = 1, while 0/0 has no value for the opposite reason: every c satisfies c · 0 = 0 and nothing chooses between them. That is why a quotient whose top and bottom are both heading to zero can still have a perfectly good limit, and why the limit has to be found some other way than by substituting. The other forms met early are infinity over infinity, zero times infinity, and infinity minus infinity.. It is not undefined for lack of a candidate. It is undefined for lack of a reason to prefer one.

So the picture has an answer and the arithmetic does not. At the exact setting where the formula gives up, the line on screen is sitting somewhere you could point at. Leave the page here able to say what is wrong, because every remaining step exists to close that gap.

Write out what the last two steps were computing. The curve was f(x) = x² and one point was held at x = 1, so the two heights are x² and 1 and the two inputs are x and 1. The slope of the line through them is (x² − 1)/(x − 1).

That expression is a rule in its own right, with x as its input, and it is the curve this step draws. The ladder of slopes you just walked down was this rule being evaluated at 0, then 0.5, 0.9, 0.99 and 0.999.

You have met it before. It is the rule the first page of the functions subject ended on, where it was left as a question rather than answered.

The bottom is 0 when x = 1, so 1 is refused and is not in the domain The set of inputs a function accepts. It is part of the function rather than a note attached to it, so two rules with the same formula and different domains are different functions: 1/x excludes the single point 0 and √x excludes every input below 0, which are two different shapes of exclusion. Sine and cosine accept every real number. Tangent and secant exclude the angles where cosine is zero, and cotangent and cosecant exclude the angles where sine is zero, because a division by zero has no value there.. Now factor the top: x² − 1 is (x − 1)(x + 1). For every other input the factor of x − 1 appears above and below and cancels, leaving x + 1. Cancelling means dividing the top and the bottom by x − 1, and that is only legal when x − 1 is not zero. So the hole survives the simplification rather than being tidied away by it. x + 1 is a different rule, one that accepts 1 and returns 2 there.

What is drawn is therefore the straight line of x + 1 with a single point taken out, marked with the open circle A small unfilled ring drawn on a number line or a curve to mark a point that is not included: the position is shown so you can see where it would be, and the ring is left hollow so you can see that no value sits there. It is the standard mark for an input outside the domain, such as x = 1 on the graph of (x² − 1)/(x − 1), which is the line x + 1 with that one point removed. A filled dot means the opposite, that the point is included. the functions subject already uses. A break of that shape has a name: a removable discontinuity A break in a function that is exactly one point wide, where the limit exists but the value is missing or disagrees with it. (x² − 1)/(x − 1) is the standard case: it returns x + 1 at every input except 1, because cancelling the common factor of x − 1 is dividing by zero there, so the graph is the line x + 1 with one point lifted out and drawn as an open circle. It is called removable because defining the value at that one input to be the limit repairs it, which is a change made to the function rather than a discovery about it. A jump or a blow-up cannot be repaired this way., one point wide, with the curve behaving normally on both sides of it.

Two facts from two different drawings are now the same fact. The slope formula had nothing to return when Δx reached 0. This curve has no height at x = 1. Same expression, same refused input, and the two steps before this one were standing on this hole the whole time without drawing it.

Step the march in before reading on. One control walks both sides inward together, so take it as far in as it goes.

From below, the rule gives 1.9 at x = 0.9, then 1.99 at 0.99, then 1.999 at 0.999. From above it gives 2.1 at 1.1, then 2.01 at 1.01, then 2.001 at 1.001. Both columns close on 2. Neither ever reaches it, because x + 1 equals 2 only at x = 1 and that is the one input the rule turns away.

That is a limit The single number the values of a function get and stay arbitrarily close to as the input is taken near a point, written lim f(x) = L as x → a. The value of f at a is not part of the statement and does not affect it: (x² − 1)/(x − 1) has no value at x = 1 and its limit there is 2, and defining the value at that one input to be anything at all would leave the limit unchanged. It exists only when the limits from the two sides both exist and agree. "Arbitrarily close" is made checkable by the epsilon-delta definition.. The limit of this rule as x goes to 1 is 2, written lim f(x) = 2 as x → 1. Every word of the statement is about the heights at inputs near 1, and not one word of it is about the height at 1.

Here is the sentence this step exists for. Bolt an extra clause onto the rule saying that at x = 1 its value is 500. The number 500 is arbitrary, and that is the point: pick any other and nothing below changes. Every row of the table stays exactly as it is, because no row in it asks about x = 1. The limit is still 2. The value at the point has no bearing on the limit, and that is the definition working as intended rather than a loophole in it.

Which is why "plug the number in" is not what finding a limit means. For a great many rules substituting the number does give the limit, and a rule for which it does is said to have continuity The property of a function at a point where three things hold: the limit exists, the value exists, and the two are equal. All three clauses are needed and they are independent. (x² − 1)/(x − 1) has a limit at x = 1 and no value there, while a function that jumps has a value at the join and no limit. Continuity is what makes substituting the number a legitimate way to find a limit, so it is a fact about the particular function rather than a general method. The picture usually offered, a curve drawn across an interval without lifting the pen, is a first idea rather than the definition: x·sin(1/x) given the value 0 at 0 is continuous on the whole of [−1, 1] and its graph has infinite length in any interval containing 0, so no pen traces it. at that point. Continuity is the coincidence of two separate things, the limit and the value, and it is a fact about the particular rule rather than a general method. Here the limit exists and the value does not, so the two come apart and you can see which one you were relying on.

Walking in from below and walking in from above are two different questions. So far they have been run together and given the same answer, which hides the fact that they are separate.

A one-sided limit A limit with the inputs restricted to one side of the point. x → a⁻ takes inputs below a and x → a⁺ takes inputs above it, and the small minus and plus name the side you arrive from rather than the sign of a number. The two-sided limit exists exactly when both one-sided limits exist and are equal, and is then their shared value, so a jump, where both are finite and different, has no limit. Sometimes the one-sided version is the only question available: √x has no inputs below 0, so at 0 there is nothing on the left to march through. is the same question with the inputs restricted to one side. Coming up from below is written x → 1⁻ and coming down from above is x → 1⁺. The small minus and plus are not signs attached to the number 1. They name which side you arrived from.

On this curve both sides settle on 2 and nothing dramatic happens. That is why the split is introduced here: get the habit of reading two numbers instead of one while the two numbers agree, so that the next step is about the disagreement and not about the notation.

The rule tying them together is short. The two-sided limit exists exactly when both one-sided limits exist and are equal, and then it is their shared value. That gives three ways to fail and the next step has one curve for each.

There is a second reason the one-sided version gets its own name. Sometimes it is the only question available. √x has no inputs at all below 0, so at 0 there is nothing to the left to march through, and the limit from the right is the whole of what can be asked.

One curve so far, and both sides of it agreed. The chooser here keeps that curve as its first option and adds three that are broken, each in a different way, so the answer is not always no and you have to look rather than guess. The two side readouts from the previous step diagnose all four, and nothing new has to be read.

A jump. The rule takes one value on the left of the point and a different one on the right. Both one-sided limits exist and both are finite, and they are not equal. There is no single number the heights are closing on, so there is no limit. This curve also has a perfectly good value at the point. It has a value and no limit, which is the exact reverse of the hole curve, which has a limit and no value. Those two questions are independent and this page is the place to stop conflating them.

A blow-up. Near the marked point the heights do not settle on anything, they grow. There is no number for them to be close to, so there is no limit, and on this one the two sides grow in opposite directions as well. People write "the limit is infinity" here and it is worth knowing what that sentence abbreviates. It says: these values grow past any bound you name. Name a million and march far enough in and the readout passes it. Name a larger one and it passes that too. Infinity is a direction the values travel, not a number they arrive at, and nothing on the curve ever equals it. The long phrasing is the one to leave with.

Something that never settles. The heights swing between fixed limits, and the swings get faster as the input closes in. March in and the readout is one number. March further and it is a different one, no closer to anything. There is no zoom level at which this resolves, and the failure is precisely that there is no last zoom level.

All three break the same claim. A limit says the heights get and stay arbitrarily close to one number. The jump gives two numbers, the blow-up gives no number, and the oscillation never stays close to anything. "Arbitrarily close" is doing all the work in that sentence and it has not yet been made into anything you can test. That is the next step.

"The heights settle on 2" was a sentence, not a test. It does not say how close, and it does not say how near the input has to be to get that close. Two people can disagree about whether a march has settled and neither of them has anything to check.

The fix is to let someone else set the standard. They name a tolerance around the settled height, called ε, the Greek letter epsilon. On the drawing that is a horizontal band, the settled height plus and minus ε. You answer with a window around the input point, of half width δ, the Greek letter delta, drawn as a vertical strip. Your answer is good if every piece of curve inside your window lies inside their band. Caught, or not caught.

Do it three times. Set a band, hunt for a window that catches the curve. Then make the band thinner and hunt again: the window you had probably no longer works, and a narrower one does. Then thinner again. The claim being made is that this never runs out, and doing it by hand three times is what makes that believable. A proof does all of them at once and is not on this page.

Now the point itself. The window contains the input point, and the definition deliberately throws that one input away. It asks about every x with 0 < |x − a| < δ, and the left half of that, the 0 <, says x is not a. Run the figure on a curve with a hole at a and the exclusion costs nothing, because there was nothing there to ask about. Run it on a curve that does have a value at a and the exclusion is what stops that one value having any say in the verdict. Both are the same clause, and it is why a limit can exist at a point where the function has no value at all.

The notation is two distances and you have already moved both of them. |x − a| is the distance from x to a on the number line, which is the reading of absolute value The size of a number with its sign discarded, written |x|. |3| and |−3| are both 3, and |0| is 0. As a function it is defined on every real number and never returns a negative one, and its graph is two straight half lines meeting at a corner at the origin. |x − a| is the distance between x and a on the number line, which is the reading that matters most later. the functions subject ended on. So the whole statement is: for every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε. That is the epsilon-delta The definition that turns "the values settle on L" into something checkable. lim f(x) = L as x → a means: for every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε. Read it as two distances. Someone names a tolerance ε around L, drawn as a horizontal band, and a window of half width δ around a must then exist that keeps every output inside that band. The order matters, because δ is allowed to depend on ε and a thinner band generally needs a narrower window. The 0 < on the left excludes x = a itself, which is why a limit can exist where the function has no value. Cauchy argued with inequalities of this kind in his Cours d'analyse of 1821, and this form settled in Weierstrass's Berlin lectures in the 1860s and 1870s. definition. Cauchy argued with inequalities of this kind in his Cours d'analyse of 1821, and the form written above settled in Weierstrass's Berlin lectures in the 1860s and 1870s.

The order of the two halves is not decoration. ε is named first and δ second, so δ is allowed to depend on ε, which is what the figure makes you do: a thinner band needs a narrower window. Write it the other way round, one window that works for every band, and you have said something false about every curve that is not flat.

A ball is dropped and you want its speed at one instant of the fall, at the moment it passes a mark on the wall. The apparatus you have is two light gates and a clock. A gate is a beam the ball breaks. The clock records the two break times, and the distance between the gates is something you measured beforehand with a tape. Nothing in that apparatus measures a speed.

What it returns is two gaps. Δs is the distance between the gates and Δt is the time between the two breaks. Divide one by the other and you have Δs/Δt, which is the average speed over the whole flight from the first gate to the second. It is the quotient from the first step of this page with metres and seconds attached to it.

You can say exactly how wrong it is. A ball released from rest has fallen ½gt² after t seconds, with g = 9.80665 metres per second per second. Over the interval from t to t + Δt the distance is ½g(t + Δt)² − ½gt², which multiplies out to g·t·Δt + ½g·Δt². Divide by Δt and the gate reading is g·t + ½g·Δt. The first part is the true speed at the first gate. The second part is the error, and it is half of g times the time the ball spent between the gates.

Read that error off the apparatus rather than trusting the algebra for it. The time between the two breaks is one of the numbers the clock hands you, so the bench reports the size of its own error. Close the gates and that time shortens, the error shrinks, and the reading closes in on the true speed. It would arrive only at no separation at all, because ½g·Δt is zero only when Δt is zero, and gates with no gap between them are not two gates.

Close them all the way and the two break times are the same time. Δs is 0, Δt is 0, and the readout is the 0/0 from the second step of this page wearing units. The speed at an instant is not a quantity this apparatus can return. It is the limit of a quantity the apparatus does return, which is a different kind of thing: not one of the readings, but the number the readings are closing on.

There is a second reason the apparatus never gets there, and it is hardware rather than mathematics. The clock has a resolution, so each break time is recorded to within that much, and the uncertainty that puts into Δs/Δt is the reading multiplied by the resolution and divided by the time between the breaks. That time is the thing you have been shrinking. The ½g·Δt error is proportional to it and the clock's contribution is inversely proportional to it, so every step down the ladder improves one and damages the other. At the finest setting the figure offers, one microsecond, a metre of spacing gives a reading uncertain by about ±0.0000286 metres per second, and a millimetre of spacing gives about ±0.0196. The two cross at around 4 millimetres of spacing, and that crossing is the best this bench can do. A coarser clock moves the crossing further out, so the best spacing is a fact about the instrument rather than about falling. It is never zero.

That generalises past this bench. Every instrument that reports a rate reports one gap divided by another, and a flow meter is the same apparatus with a counted volume where the gate spacing is. The rate at an instant is what a sequence of those measurements is heading for, and it is the one value none of them ever hands back.

Put the two side by side, because they look alike and are not alike at all.

Δx is a number. Set it to 0.1 and you can point at a bracket on the drawing that is 0.1 wide. Δy is a number in the same way. Δy/Δx is a division of two numbers, it is the tilt of a line through two genuinely separate points, and every part of it is ordinary arithmetic.

Run the collapse. As Δx goes to 0 the two points become one, the line steadies, and the quotient stops being a division: top and bottom both reach zero and 0/0 chooses nothing. What survives is not the quotient but the limit of the quotient, and by everything on this page that limit is a number and can be checked. The notation for it is dy/dx.

Here is the misreading, and it is the one this page was built to prevent. dx looks like a very small number. It is not a number at all, and there is no value you can give it. Set Δx to 10⁻³ and it is 0.001. Imagine a rung at 10⁻⁶ and it would be 0.000001. Neither of those is dx. They are two rungs of a ladder, and dx is not the last rung, because there is no last rung. A dx The dx and dy in dy/dx. Neither is a number. dx is what the notation keeps of Δx once the limit Δx → 0 has been taken, and there is no value you can give it: 0.001 and 0.000001 are small values of Δx, and neither is dx, because dx is not the last rung of that ladder and the ladder has no last rung. So dy/dx is a single symbol for the limit of Δy/Δx, not a fraction whose top and bottom may be separated, and the bar in it is spelling inherited from the division that was real before the limit was taken. Leibniz chose the notation in the 1670s and 1680s because several manipulations that look like cancelling do give correct answers, but each of those is a theorem proved separately rather than the rules of fractions applying. is what the notation keeps once the whole ladder has been replaced by the number the ladder was heading for.

The consequence follows immediately. dy/dx is a single symbol. It is not a fraction with a top and a bottom you may separate, because neither dy nor dx names a number on its own. The bar in it is inherited spelling from the Δy/Δx that really was a division before the limit was taken. Leibniz chose it that way in the 1670s and 1680s on purpose, because several manipulations that look like cancelling do give correct answers. Each of those is a theorem that gets proved separately, and none of them is the rules of fractions applying.

Reverse the collapse, and this is the part worth doing rather than reading. Δx comes back with a size, and dx does not grow back into one, because it never was one. Running it both ways is what shows the two are different kinds of object rather than the same object at two scales.

One thing has deliberately gone unnamed. The settled slope, the thing the line was sitting at when the formula gave up in the second step, has a name and this page has not given it. What the page has built instead is the only machinery that can carry it: the limit of Δy/Δx as Δx goes to 0, which did not exist before the fourth step and could not be checked before the seventh. The name will be the next page's opening line.

Every number on this page is worked out from the controls as you set them, including the ones in the legends, the tables and the arithmetic panels. Nothing here was stored as a picture. The falling ball is integrated one clock tick at a time and never reads the formula for a falling body, so the dashed line it walks towards is a second calculation and not the same one drawn twice. No figure shares a control with any other: each one starts fresh where its own step needs it to start.