The limit, seen as a game between ε and δ

How to use this simulation

1. Pick a function under “Function f(x)”. The thick curve is y = f(x). The vertical dotted line marks the point a, which you can drag along the graph or set with “Point a”.
2. The orange horizontal band is the ε band: the strip of heights within ε of the limit value L. “ε” is the error you are willing to put up with in the answer. Change it with the “ε” slider.
3. The green vertical band is the δ band: the widest strip of x-values around a for which the curve, apart from the single point x = a itself, stays inside the orange band. The readout “Widest δ that works” gives its half-width.
4. Press play. ε shrinks again and again, and the green band narrows with it. That is the whole definition in motion: whatever ε you choose, a δ exists.
5. Read “Left-hand limit” and “Right-hand limit”. When they agree the limit exists and is shown as L. The hollow circle on the curve marks a point where the function itself has no value.
6. Choose the last function, the jump. Put a at 0. The two sides disagree, and for any ε of 1 or less the green band disappears with the message “no δ works”.

What does “approaching” really mean?

Say you are walking towards a shop. You get closer, then closer again. “Approaching” is about the journey, not the arrival. A limit asks the same thing of a function: as x creeps towards a number a, what height does f(x) creep towards? We never ask what happens at a itself, only near it.

That distinction matters. Take (x² − 1)/(x − 1). At x = 1 it is 0 ÷ 0, which is meaningless, so the function has no value there at all. But for every other x it behaves perfectly well. The limit lets us say something useful about a place the function itself refuses to answer.

The ε–δ game

Textbooks write the definition with Greek letters and it looks scary. It is really a game between two people. You are the sceptic and you pick a tolerance ε, a tiny positive number: “I want f(x) within ε of L.” The other player has to answer with a δ: “then keep x within δ of a, and I promise you will get it.”

If the second player can answer every ε you throw, however small, the limit is L. If there is even one ε for which no δ works, the claim is false. In the simulation the orange band is your demand and the green band is the answer.

• 0 < |x − a| means x is near a but not equal to a. The point x = a is deliberately left out.
• |f(x) − L| < ε means the height is inside the orange band.
• δ is allowed to depend on ε. A tighter ε usually needs a smaller δ.

A straight line: δ is just ε divided by the slope

Start with f(x) = 2x − 1 at a = 1, where L = 1. Moving x by δ moves f(x) by twice as much, so the output error is 2δ. To keep 2δ below ε we need δ = ε/2. The table below was computed by the simulation's own search, and it matches.

A curve: x² bends the rules a little

Now f(x) = x² at a = 1, with L = 1. The curve is steeper on the right of 1 than on the left, so the two sides allow different widths and the safe δ is the smaller one. The simulation shows the green band sitting evenly around a, sized by the tighter side.

• Shrink ε by 10 and δ shrinks by roughly 10 as well, once ε is small. Near a point a smooth curve looks like its tangent, so δ ≈ ε / slope again.
• The same ε needs a different δ at a different place. At a = 2, where the slope is 4 and L = 4, ε = 0.5 gives δ = 0.1213.

A hole: the limit exists, the value does not

Choose (x² − 1)/(x − 1). For every x except 1 this equals x + 1, so the graph is the straight line y = x + 1 with a single point missing, drawn as a hollow circle. As x closes in on 1 from either side, the heights close in on 2.

• The limit is 2, even though f(1) is undefined.
• Here the green band is as wide as ε itself, since the line has slope 1. For ε = 0.5, 0.1 and 0.01 the widest δ values are 0.5000, 0.1000, 0.0100.
• This kind of gap is called a removable discontinuity: define f(1) = 2 and the hole is patched.

sin x / x at 0: the most famous limit

Pick sin x / x and leave a at 0. At x = 0 it is 0 ÷ 0 again, and the curve has a hollow circle. Yet look at the values on both sides: they crowd towards 1.

A jump: when no δ works

Choose the jump function: it equals -1 for every negative x and 1 from zero onwards. Put a at 0. Coming from the left the heights head for -1; from the right, for 1. The two one-sided limits disagree, so there is no single L.

The simulation still draws a band around the midpoint 0 so you can try. For ε up to 1 the curve always pokes out of the band, however close to 0 you stay, and the table shows it.

• ε = 0.5 and 0.9: δ = 0. No width saves you, because both pieces sit a full unit from the midpoint.
• ε = 1.1: every δ works, since the whole curve lies within 1.1 of 0. Such a loose demand proves nothing. A limit must survive every ε, including the tiny ones.
• Away from the jump, say at a = 2, the two sides agree again (both 1) and the limit exists.

Left, right, and when a limit exists

The left-hand limit uses only x < a and the right-hand limit only x > a. The ordinary limit exists exactly when both exist and are equal. That is the quickest test in practice: compute the two sides, compare.

Continuity is the stronger claim. A function is continuous at a when the limit exists, the function has a value at a, and the two are equal. A hole breaks the second part, a jump breaks the first, and both the hole and jump functions in the simulation are therefore discontinuous at their special points.

Try it in the simulation

Six quick experiments, a minute each.

• Choose 2x − 1 and press play. Watch the green band stay exactly half as wide as the orange one.
• Pause, drag a along x² to a large value like 2.5. Does the same ε still give the same δ?
• Choose the hole function and drag a to 1. See the hollow circle. Now drag a to 2: the circle turns into a solid dot.
• Choose sin x / x and use a large ε of 1.5. The green band becomes enormous. Shrink ε and watch it close in around 0.
• Choose the jump, a = 0, and set ε to 0.5. The message “no δ works” appears. Raise ε above 1. Why does the band come back?
• On the jump, drag a to 1. The limit now exists. How wide can δ be before it reaches the jump at 0?

Solved problems

Six common questions. Every number below is computed with the same code as the simulation.

Mistakes to avoid

These cost marks again and again.

• Thinking the limit is f(a). The limit ignores what happens at a; for the hole and sin x / x functions f(a) does not even exist.
• Writing a limit when the two sides differ. Always check both sides if the function changes rule at a.
• Choosing δ first. In the definition ε comes first and δ is found from it.
• Cancelling a factor that is zero at the very point and forgetting that the original function is still undefined there.
• Writing = 0 ÷ 0 = 1. 0 ÷ 0 is meaningless. It is a signal to factorise or rearrange, not an answer.

Where limits are used

Every derivative is a limit (the slope of a secant as the gap shrinks to zero), and every integral is a limit (the sum of thin strips as the strips narrow). Without limits there is no calculus.

Engineers use the same idea in tolerance: a part has to be within ε of a specified size, and the question is how well the machine must be set, which is δ. Computer scientists use it for convergence: how many steps until the answer is within ε.

Exam corner

For HSC higher maths and admission tests, remember these.

• Try substitution first. If you get a number, that is the limit. If you get 0/0, factorise, rationalise or use a standard limit.
• The standard limit lim (x → 0) sin x / x = 1 is used constantly, with x in radians.
• To show a limit does not exist, find two different one-sided limits, or two paths with different answers.
• In a formal ε–δ proof, work backwards from |f(x) − L| < ε to find what |x − a| must be, then write the proof forwards.

Frequently asked questions