You're posting a parcel. The price list says: up to 100 g — £1.55; over 100 g — £2.70. Now imagine loading the parcel gram by gram and watching the price. Creep up towards 100 g from below — 99 g, 99.9 g, 99.99 g — and the price sits stubbornly at £1.55 the whole way. Creep down towards 100 g from above — 101 g, 100.1 g, 100.01 g — and it sits just as stubbornly at £2.70. Same target weight, two completely different destinations, and the only difference is which direction you came from.

Nature does it too. Cool a glass of water down towards 0\,^\circ\text{C} and at every temperature just above zero you're still holding liquid. Warm a block of ice up towards 0\,^\circ\text{C} and at every temperature just below zero it's still solid. Approach the very same temperature from the two sides and the world looks different — liquid one way, ice the other.

When we found a limit, we sneaked up on c from both directions at once and the outputs met at one value. The parcel and the ice show that sometimes they don't meet: the approach from the left and the approach from the right tell different stories. Mathematics needs a way to talk about each story separately — and that is exactly what one-sided limits are for.

A name for each direction

We bolt a tiny superscript onto the c to say which side we're approaching from:

• The left-hand limit — approach with x a little below c (through values like c - 0.1,\ c - 0.01,\ c - 0.001, \ldots): \lim_{x \to c^-} f(x).
• The right-hand limit — approach with x a little above c (through values like c + 0.1,\ c + 0.01,\ c + 0.001, \ldots): \lim_{x \to c^+} f(x).

Read them aloud as "the limit of f(x) as x tends to c from below" and "…from above". The little {}^- and {}^+ are like arrows pointing at c from each side: minus for the side where x is less than c, plus for the side where it is greater. Everything else works exactly as for ordinary limits — we watch where the outputs are heading, and we never actually land on c itself.

A step in the road

Picture a function that jumps at x = 2: it sits at height 1 just to the left, then leaps up to height 3 just to the right — a mathematical version of the parcel price. Use the slider to walk a point in from each side and read where it lands.

Do it slowly, and keep a diary of the approach — the same numeric squeeze you used for two-sided limits, but run one side at a time:

From the left the point heads for 1; from the right it heads for 3: \lim_{x \to 2^-} f(x) = 1, \qquad \lim_{x \to 2^+} f(x) = 3. Two clean, confident answers — one for each side. Notice that each one-sided limit is a perfectly respectable limit in its own right: the left-hand values genuinely settle on 1, the right-hand values genuinely settle on 3. Nothing has "failed" yet. The trouble only starts when we ask the two sides to agree.

The two-sided limit needs agreement

Here is the rule that ties everything together — and it's an if and only if, which makes it doubly useful: it tells you when a limit exists and gives you a foolproof way to prove one doesn't.

For our jump, the sides disagree (1 \ne 3), so the two-sided limit at x = 2 does not exist — even though each one-sided limit is perfectly fine on its own. That is the standard way to demolish a limit: compute the two sides separately and show they clash.

One more place one-sided limits earn their keep: the edge of a domain. Take f(x) = \sqrt{x} at x = 0. There is simply no function to the left of zero — \sqrt{x} isn't defined for negative x — so a left-hand limit is meaningless. But the right-hand limit is perfectly sensible: \lim_{x \to 0^+} \sqrt{x} = 0. Whenever a function lives on only one side of a point, the one-sided limit is the only limit you can ask for there.

Worked example: a piecewise function, side by side

Piecewise functions are where one-sided limits become a routine tool, because the formula itself changes at the joint. Take

and ask for \lim_{x \to 2} f(x). The strategy is always the same three moves.

Move 1 — the left-hand limit. Approaching from the left means x < 2, so only the top branch applies. That branch is a harmless polynomial, so its limit is found by substituting: \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x + 1) = 2 + 1 = 3.

Move 2 — the right-hand limit. Approaching from the right means x > 2, so only the bottom branch applies: \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (7 - 2x) = 7 - 4 = 3.

Move 3 — compare. Left gives 3, right gives 3. They agree, so by the theorem the two-sided limit exists: \lim_{x \to 2} f(x) = 3. The two branches are different formulas, but at the joint they happen to land on the same value — the graph's two pieces meet, and there is no jump at all.

Now tweak one number. Replace the bottom branch with 8 - 2x and rerun the three moves: the left-hand limit is still 3, but the right-hand limit becomes 8 - 4 = 4. Now 3 \ne 4, the pieces miss each other, and \lim_{x \to 2} f(x) does not exist. One digit in the formula is the difference between a seamless join and a jump — and the three-move check detects it every time.

Worked example: the classic \dfrac{|x|}{x}

Here is the most famous jump in all of calculus. Let g(x) = \dfrac{|x|}{x} and ask what happens at x = 0. First note g(0) itself is undefined — it would be \tfrac{0}{0} — but limits never cared about the point itself, so we press on.

To the right of zero, x is positive, so |x| = x and the fraction collapses: x > 0: \quad g(x) = \frac{|x|}{x} = \frac{x}{x} = 1. Every single output on the right is exactly 1 — try g(0.5) = 1, g(0.001) = 1. So \lim_{x \to 0^+} g(x) = 1.

To the left of zero, x is negative, so the absolute value flips the sign: |x| = -x. Now x < 0: \quad g(x) = \frac{-x}{x} = -1, and every output on the left is exactly -1. So \lim_{x \to 0^-} g(x) = -1.

Left limit -1, right limit +1: they disagree, so \lim_{x \to 0} g(x) does not exist. This little function — sometimes called the sign function, since it reports the sign of its input — is worth memorising: it is the standard example that both one-sided limits can exist beautifully while the two-sided limit fails completely.

Worked example: reading a staircase off a graph

The parcel-price idea has an official mathematical mascot: the floor function \lfloor x \rfloor, which rounds down to the nearest whole number. So \lfloor 2.7 \rfloor = 2, \lfloor 2.99 \rfloor = 2, and then, the instant you reach 3, \lfloor 3 \rfloor = 3. Its graph is a staircase — every price band, tax band and "round down to the minute" billing rule in the real world has this shape.

Read the one-sided limits at x = 3 straight off the picture. Coming from the left you're walking along the tread at height 2 (the outputs at 2.9, 2.99, 2.999 are all 2), so \lim_{x \to 3^-} \lfloor x \rfloor = 2. Coming from the right you're on the tread at height 3, so \lim_{x \to 3^+} \lfloor x \rfloor = 3. The sides disagree, so the two-sided limit at 3 does not exist — and the same argument works at every integer. Between the integers, though, both one-sided limits agree (at x = 2.5 both sides give 2), and the limit exists happily.

Notice the dots. The filled dot at (3, 3) says the function's value there is 3; the open circle at the right end of each tread says the tread does not include its endpoint. The left-hand limit at 3 is 2 regardless of the filled dot — a one-sided limit only ever looks at the approach, never at the arrival.

Watch Sal read them off a graph

Reading one-sided limits from a graph — following each side's approach with your finger and ignoring any stray dots — is a skill worth drilling. Watch it done slowly: