Limits

A limit is the number a function’s values approach as its input approaches some point. The function does not have to reach that number, and it does not even have to be defined at the point. The limit describes where the values are heading, not where they end up. Every idea in calculus, from the slope of a curve to the area under it, is a limit of this kind.

Need a number now? The limit calculator gives both one-sided limits of any function you type, with error estimates and the values it used.

On the home page you watched a secant line turn into a tangent as the gap h shrank to nothing. But at h = 0 the slope formula divides zero by zero, which means nothing at all. Calculus works because we never set h to zero. We ask what the slope is heading toward as h gets small. Making that question precise took mathematicians about 150 years. You can see the whole idea in a few minutes.

Approaching a point you cannot stand on§

Take the function f(x) = (x² − 1)/(x − 1). At x = 1 it is 0/0: undefined. Try a few inputs near 1 in your head, or just guess, before you look.

The algebra agrees with what the probes showed. Everywhere except x = 1 you can cancel the common factor, because x² − 1 = (x − 1)(x + 1), and the function is x + 1. Its graph is a straight line with one point missing, and the values on both sides crowd toward 2.

Push the distance slider all the way. At a distance of 0.0001 from 1, the two probes read 1.9999 and 2.0001. They are pinching the number 2 from both sides, and they would keep pinching it however close you went. That is what we write as

and read as “the limit of (x² − 1)/(x − 1) as x approaches 1 is 2”^[1]. Two features of this definition surprise people, and both matter.

First, the value at the point is irrelevant. The limit is built from nearby inputs only. That is why it can exist where the function is undefined, and why it can differ from the function’s value. Choose the third function in the figure: it is x² everywhere except at x = 1, where someone has moved the point up to 4. The limit is still 1, the height the curve is heading for, and the stray point is ignored. A function whose limit at every point equals its value there is called continuous; that is the precise version of “you can draw it without lifting your pen.”

Second, a limit is a number, not a process. It is tempting to think of a limit as the function “moving toward” a value and never quite arriving. Students hold onto this picture with remarkable tenacity. Steven Williams found that this dynamic view resisted change even after students met counterexamples to it^[2], and David Tall and Shlomo Vinner showed that the private mental picture of a limit that students build often conflicts with the formal definition they are taught^[3]. The motion is a fine way to find a limit. But the limit itself is the fixed number 2, and the constant function f(x) = 5 has limit 5 everywhere without moving at all.

The second example in the figure, sin x / x at 0, is the famous one. There is no factor to cancel, and yet the probes close in on 1. This limit is the reason the derivative of sin x is cos x, and it is the reason calculus measures angles in radians: in degrees the limit would be π/180 instead of 1.

Left, right, and when the two disagree§

The probes in the figure approach from both sides at once. You can also approach from one side only. The left-hand limit, written with x → a⁻, uses inputs below a; the right-hand limit, x → a⁺, uses inputs above. The ordinary two-sided limit exists exactly when both one-sided limits exist and are equal^[1].

The floor function shows the simplest failure. Approach 2 from the left and the values sit at 1 the whole way; approach from the right and they sit at 2. Each one-sided limit exists. They disagree, so the two-sided limit does not. This is not a technicality. A jump like this is exactly where a rate of change stops making sense, and the next chapter will meet it again as a function with no derivative.

The second failure is a pole. As x approaches 1, 1/(x − 1)² grows past every number you name: past 100 at a distance of 0.1, past a million at 0.001. We write the limit as +∞, but that is shorthand for “there is no limit, and here is the particular way it fails”^[1]. Infinity is not a number the values approach; it is a description of their growing without bound.

The third failure is the strangest, and it gets its own section below.

Making “close” precise: the ε–δ game§

“Approach,” “close” and “heading toward” are pictures, not definitions. For a century and a half after Newton and Leibniz, calculus ran on pictures like these, and on an even shakier idea: the infinitesimal, a quantity smaller than any positive number yet not zero. The results were right, but nobody could say what the objects were. In 1734 the philosopher George Berkeley attacked the whole enterprise in The Analyst, asking whether Newton’s vanishing increments might be called “the ghosts of departed quantities”^[4].

The answer that finally satisfied mathematicians replaces motion with a challenge and a response. Suppose I claim the limit of f at a is L. You challenge me with a tolerance ε (epsilon): any positive number, as small as you like. I must answer with a distance δ (delta) such that every input within δ of a, other than a itself, gives an output within ε of L. If I can always answer, however small your ε, the limit is L^[1]:

Play a round with x² near a = 2, where the limit is 4. Set ε = 0.5. The figure finds δ ≈ 0.1213: every x between 1.8787 and 2.1213 has x² between 3.5 and 4.5. (The exact answer is δ = √4.5 − 2, because the curve is steeper on the right and leaves the band there first.) Shrink ε to 0.01 and δ shrinks to about 0.0025. It always shrinks, but it never hits zero, so I can always answer. That is the whole definition at work.

Now claim the limit is 4.3 instead. For a generous ε the game still works. But as soon as ε is smaller than 0.3, the band no longer contains 4, and points arbitrarily close to 2 escape it. No δ works, and the claim fails. The definition distinguishes the true limit from every impostor, which is what a definition is for.

Try the floor function at 1. Whatever L you claim, any band narrower than the jump leaves one side or the other outside, so no limit exists. And try x sin(1/x) at 0: it wiggles infinitely often, yet δ = ε works, because the wiggles are trapped between the lines y = x and y = −x.

documented

A function that never settles§

The function sin(1/x) is defined everywhere except 0. As x shrinks toward 0, 1/x races off to infinity, and the sine of it sweeps between −1 and 1 faster and faster. Between 0.1 and 0.01 alone it completes about 14 full oscillations; between 0.01 and 0.001, about 143.

This is the third way a limit can fail: no jump and no blow-up, but no settling either^[1]. Multiply by x and the picture changes completely. The function x sin(1/x) oscillates just as often, but its swings are bounded by |x|, which goes to 0. Trapped between −|x| and |x|, it has no choice: its limit at 0 is 0. This argument has a name, the squeeze theorem, and it is how the limit of sin x / x is proved.

The wiggling function carries a warning about computation too. If you evaluated sin(1/x) only at x = 1/π, 1/(2π), 1/(3π), and so on, you would get exactly 0 every time and conclude, wrongly, that the limit is 0. A table of values is evidence, never proof, and a function can hide anything between the rows.

Finding limits in practice, and a calculator that shows its evidence§

In practice limits are found with a short list of moves^[1]. If the function is continuous at a (polynomials, sines, exponentials and anything built from them by arithmetic and composition, away from division by zero), substitute: the limit is the value. If substitution gives 0/0, simplify first: factor and cancel, as in equation (1.1), or rationalize a square root. If that fails, squeeze it, or use one of a few standard limits such as sin x/x → 1. And if the two sides disagree, the limit does not exist.

The calculator below takes a different, numerical route, so you can check your work and see the evidence. It evaluates your function at inputs ever closer to a on each side, then extrapolates the trend to distance zero. That extrapolation is powerful: for (1 − cos x)/x² it finds 0.5 to about ten digits, even though evaluating the formula directly at x = 10⁻⁸ gives 0 because the computer runs out of digits. It also reports, honestly, when the values do not settle.

Try (1+x)^(1/x) at 0. The calculator returns 2.718281828…, the number e, which can be defined as exactly this limit. Try 1/x at 0, and it tells you the two sides run off in opposite directions. And try x^x at 0: from the right it heads to 1, and from the left the function is undefined, so only a one-sided limit exists.

Why calculus needs limits§

Everything that follows on this site is a limit in disguise. The derivative is the limit of slopes of secant lines as the second point slides into the first:

At h = 0 this is 0/0, exactly the situation of equation (1.1), and it is the reason we spent this chapter on holes. The integral is the limit of sums of thin rectangles as their number grows without bound. Neither would make sense without a definition of limit that does not depend on motion, infinitesimals or ghosts.

The next chapter takes equation (1.3) and puts it to work: what the derivative is, how to read it from a graph, why the familiar rules are true, and how a computer evaluates it without ever setting h to zero.