Calculus I › Limits › full formula sheet
The constant multiple law
Constants pull straight out of limits — but only genuine constants.
On this page c is a genuine constant (no x inside) and L = limx→a f(x) exists (finite). The most abused law in the chapter is abused exactly here.
Notation in this lesson
- c
- a genuine constant — no x inside
- L
- limx→a f(x), assumed to exist and be finite
- limx→a
- the limit as x approaches a
Where it comes from
The simplest limit law — and the most abused. Scaling a graph vertically by c scales every limit by c: if f(x) hugs L, then 5·f(x) hugs 5L. One line, enormous leverage: it is what lets you pull coefficients out before thinking.
The abuse is “pulling out” something that isn’t constant:
The intuition is scaling: multiplying every function value by c multiplies every error by |c|, so an error budget of epsilon for f becomes |c|·epsilon for c·f — still arbitrarily small. The derivation is two lines.
Derivation
Two routes — the one-line route and the epsilon-delta route. Both are worth seeing.
Key steps shown; both arguments are complete. Route A shows the law is redundant given the product law — it survives as a separate law purely because you use it a hundred times a day. The difference law’s proof (f − g = f + (−g)) leans on this law for the −1.
How to use it
The procedure, every time:
- Confirm c is genuinely constant. No x inside — not x², not sin x, not 1/x. Parameters (k, π, a) that don’t depend on x are fine.
- Pull it out, evaluate lim f, then multiply back.
- Combine freely with sum/product/quotient laws — pulling constants out first usually simplifies everything downstream.
Constants vs parameters vs variables
In lim(x→a) k·x², k pulls out whether k is a number (5), a named constant (π), or a parameter — what matters is that k doesn’t contain x. But c may depend on a (the point you’re approaching): in lim(x→a) a·x, the “a” is constant with respect to x, so it pulls out to give a².
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: lim(x→3) 5x²
- Confirm: 5 is constant. Pull it out: 5 · lim(x→3) x².
- Evaluate: lim x² = 9 (power law).
- Multiply back: 5 · 9 = 45.
Your turn: limx→4 3x² = ?
Answer: 48.
Pull out the 3: 3·limx→4 x². Then lim x² = 16, so 3·16 = 48.
Example 2 — negative constant: lim(x→0) −2 sin x
- Pull out −2: −2 · lim(x→0) sin x.
- Evaluate: sin is continuous, sin 0 = 0.
- Multiply back: −2 · 0 = 0.
Your turn: limx→0 4 cos x = ?
Answer: 4.
Pull out the 4: 4·limx→0 cos x. Cosine is continuous and cos 0 = 1, so 4·1 = 4.
Example 3 — combined with the sum law: lim(x→1) (7x³ − 2x)
- Split first (sum law): lim 7x³ − lim 2x.
- Pull out each constant: 7 · lim x³ − 2 · lim x = 7·1 − 2·1.
- Combine: 7 − 2 = 5.
Your turn: limx→2 (3x² + 5x) = ?
Answer: 22.
Split (sum law), then pull each constant: 3·lim x² + 5·lim x = 3·4 + 5·2 = 12 + 10 = 22.
Example 4 — the trap: lim(x→0) x·(1/x)
- Can we pull out 1/x? No — it contains x (and its limit DNE). The law is off.
- Can we pull out x? Also no — x is the variable.
- Simplify instead: x·(1/x) = 1 for x ≠ 0, so the limit is 1.
Your turn: limx→0 x²·(1/x) = ?
Answer: 0.
Neither factor is constant (both contain x), so the law is off. Simplify first: x²·(1/x) = x for x ≠ 0, so the limit is 0.
Memorization tips
- The x-ray test: cover the factor — if you can still see an x, it doesn’t pull out. 5, π, k, −2 pull out; x, 1/x, sin x stay in.
- Pull out early: constants pulled out before splitting usually make every downstream step simpler. It’s the cheapest simplification in the chapter.
- Signs travel with the constant: pull out (−2), not 2-with-a-minus-to-remember-later.
- One-line proof as memory hook: constant function g(x) = c has limit c; product law does the rest. If you forget the law, rebuild it in five seconds.
- c may depend on a: in lim(x→a), anything without x is constant — even the letter a itself.
- The law that powers the difference law: f − g = f + (−1)·g. The −1 pulls out by exactly this law.
Final challenge
Five mixed questions — basics, applications, and the traps, all in one. Score 5/5 and this law is yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the constant multiple law for limits?
If lim(x→a) f(x) = L exists and c is constant, then lim(x→a) c·f(x) = c·L: genuine constants pull straight out of limits.
Can I pull out 1/x or sin x?
No — the law only applies to genuine constants (no x inside). Pulling out 1/x from lim(x→0) x·(1/x) is illegal; simplifying first gives the correct answer, 1.
Why does the proof work?
Two views: (A) the constant function g(x) = c has limit c, so the product law gives c·L in one line; (B) |cf−cL| = |c|·|f−L|, so rescaling the error budget by |c| lands exactly on ε.
Can the constant depend on a, the point I'm approaching?
Yes — 'constant' means constant with respect to x. In lim(x→a) a·x, the a pulls out (giving a²) because a doesn't vary with x.
How does this law relate to the difference law?
The difference law is proved by writing f−g = f+(−1)·g: the −1 pulls out by the constant multiple law, and the sum law finishes.
More from the codex
Support the codex
This page is free, with no account and no ads. If it helped you learn, consider supporting the indie dev behind it.
Questions or a bug to report? Email [email protected].