When you first met the idea of a limit, you caught it by sneaking up on it: a table of inputs creeping toward c, a column of outputs closing in on a target. That worked — but picture doing it for every limit you ever need. A fresh table for x^2 + 3x, another for \tfrac{x^2+1}{x+2}, another for every polynomial in every exam question, forever. Nobody does calculus that way, for the same reason nobody works out 47 \times 23 by counting on their fingers: there are rules.

Here is the beautiful fact this page is about: limits respect arithmetic. If you know the limits of two simple pieces, you automatically know the limit of their sum, their difference, their product, their quotient. A complicated expression snaps apart into simple bricks — like a LEGO model — you take the limit of each brick (trivial), and then the same laws snap the answers back together. Five short algebra rules replace an infinity of tables. This is the moment limits stop being a curiosity and become a usable tool.

Throughout the page, suppose both of these limits exist:

Then, in one slogan: the limit of a combination is the same combination of the limits.

Two atoms and five laws

Every polynomial — in fact every rational function — is assembled from just two kinds of atom: constants and the variable x itself, glued together with +, -, \times and \div. The atoms have the two easiest limits in mathematics:

A constant doesn't care where x is heading — it just sits there at k. And the function x goes exactly where x goes. So if the limit can pass through every +, \times and \div in an expression, it will eventually reach the atoms — and the atoms are trivial. That "passing through" is precisely what the laws license:

• Sum / difference: \lim (f \pm g) = L \pm M — the limit passes through a + or -.
• Constant multiple: \lim (k\,f) = k\,L — pull constants straight out front.
• Product: \lim (f \cdot g) = L \cdot M — and since f^2 = f \cdot f, this gives the power rule \lim f^{\,n} = L^{\,n} for free.
• Quotient: \lim \dfrac{f}{g} = \dfrac{L}{M} — but only when M \ne 0. This is the one law with a tripwire; more on it below.

Read the hypotheses as carefully as the conclusions. Each law asks for two things before it will work for you: both ingredient limits must exist (be actual finite numbers), and the quotient law additionally demands a nonzero denominator limit. When the conditions hold, the law guarantees the combined limit exists and tells you its value. When they don't hold, the law doesn't say the limit fails — it simply says nothing at all, and you must reach for other tools.

The sum law, in pictures

Why should \lim(f + g) = L + M be true? Picture the two graphs near x = c. Saying f(x) \to L means that close to c, the value f(x) is hugging L — within a whisker of it. Likewise g(x) is within a whisker of M. Add the two values and the errors add too: f(x) + g(x) is within two whiskers of L + M. Shrink the whiskers and the sum is trapped — it has nowhere to go but L + M. (That two-whisker squeeze is the formal proof, dressed in plain clothes.)

Watch it happen. Below, the first curve is f, the second is g, and the third is their sum. Slide x toward c = 1 and read the panel: as f settles onto 2 and g settles onto 1, the sum has no choice but to settle onto 3. The third height is always exactly the first two stacked.

Approach c = 1 from the left, then from the right — the stacking argument doesn't care which side you come from, which is exactly why the law holds for the full two-sided limit.

Worked example 1: a polynomial, dismantled

Let's use the laws at full pedantic speed once, citing every move — the way you'd show a proof examiner. Find

No table, no graph — just take the expression apart brick by brick:

Every single step is one law. Notice where the expression finally touched ground: the only limits actually computed were the two atoms, \lim x = 2 and \lim 1 = 1. The laws did all the carrying. And notice the punchline hiding in the answer: 13 is exactly what you get by substituting x = 2 into 2x^2 + 3x - 1. That is no coincidence — it's a preview of limits by substitution, and the laws are the reason it works.

Worked example 2: chaining laws with given limits

Exam questions love to hand you limits ready-made and ask you to combine them — no formulas for f and g at all. The laws don't care: they run on the values L and M, not on formulas. Suppose \lim_{x\to c} f = 4 and \lim_{x\to c} g = 3. Then:

Constant multiple, then sum, then power — each law applied in turn, and at no point did we need to know what f and g actually are. The reference curves below show one concrete pair of functions and their sum, so you can see with your eyes that the heights of the two thin curves really do add up to the bold one, everywhere — the sum law is just this picture read at a single x.

Worked example 3: division — check the bottom first

The quotient law comes with a pre-flight check. Before you split a fraction's limit into top-over-bottom, you must confirm the bottom's limit isn't zero. Find

Step 1 — the bottom. By the sum law and the atoms, \lim_{x \to 3}(x + 2) = 3 + 2 = 5. That's not zero, so the quotient law is cleared for use.

Step 2 — the top. Power law plus sum law: \lim_{x \to 3}(x^2 + 1) = 3^2 + 1 = 10.

Step 3 — divide the limits.

Three lines, done. Get into the habit now: whenever a limit has a fraction in it, your first move is to take the bottom's limit on its own. If it's nonzero, the quotient law hands you the answer. If it's zero… read on.

When the quotient law goes silent

Now try the same routine on a fraction that fights back:

The pre-flight check fails. The bottom's limit is \lim_{x \to 2}(x - 2) = 0. The quotient law's condition M \ne 0 is violated, so the law refuses to answer. Careful with what that means: it does not mean the limit is undefined, or infinite, or zero. The law is simply silent — like asking a train timetable about a bus route. The limit might still exist; the law just can't see it.

In fact, look at the top: \lim_{x \to 2}(x^2 - 4) = 0 as well. Both storeys of the fraction are collapsing to zero at once — the notorious \tfrac{0}{0} stand-off, where the answer could be literally anything until you do more work. The work here is algebra before limits: factor the top.

Cancelling (x - 2) is legal precisely because a limit as x \to 2 never consults x = 2 itself — only the values nearby, where x - 2 \ne 0. On that neighbourhood our fraction and x + 2 are the same function, so they must share the same limit. And x + 2 is a job for the sum law:

So the limit existed all along — the quotient law just wasn't the right witness. This "denominator dies, factor first" situation has a whole theory of its own; you'll meet it properly in indeterminate limits. For now the takeaway is the drill: bottom limit zero → quotient law off the table → reach for algebra.

Watch Sal walk through them

Sal Khan runs through the same properties with a worked example per law — a good second pass if you like hearing the rules narrated over the algebra.