Calculus I › Limits › full formula sheet
The power law for limits
Evaluate the inside, then raise to the power — for positive integer exponents.
On this page n is a positive integer (1, 2, 3, …) and L = limx→a f(x) exists (finite). Fractional or negative exponents play by stricter rules — see below.
Before this lesson: The product law
Notation in this lesson
- [f(x)]n
- the nth power of f
- Ln
- the limit raised to the n
- n
- a positive integer here
Where it comes from
The problem: limits like lim(x→2) (x²+1)³ come up constantly — a whole expression raised to a power. Multiplying it out as (x²+1)(x²+1)(x²+1) and splitting three ways works, but the power law packages that tedium into one move: evaluate the inside (5), then cube it (125).
The naive overreach is assuming this works for any exponent. It does not:
The intuition is pure product law: [f(x)]n is f(x) multiplied by itself n times, so its limit is L multiplied by itself n times. The derivation is that sentence, formalized as induction.
Derivation
We prove: if lim(x→a) f(x) = L, then lim(x→a) [f(x)]n = Ln for every positive integer n — by induction on n.
Key steps shown; the induction is complete. Why does induction guarantee every intermediate limit exists? Because each step’s hypotheses are exactly the previous step’s conclusion — the product law is never invoked on a divergent piece. For n = 0 the formula would give [f]0 = 1 (where defined), consistent but usually stated for n ≥ 1.
How to use it
The procedure, every time:
- Confirm the exponent is a positive integer. Fractional exponents → root law with domain checks; negative integer exponents → quotient law (needs L ≠ 0).
- Evaluate the inner limit L = lim(x→a) f(x), using whatever tools fit.
- Raise: the answer is Ln.
- Nested with other laws? Work inside-out: innermost limit first, then apply the power, then whatever is outside.
Power law vs expanding
For lim(x→2) (x+1)² you could expand to x² + 2x + 1 and split three ways — or evaluate the inside (3) and square (9). Same answer; the power law skips the algebra. Reach for it whenever the inside is simpler than the expansion.
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→2) (x²+1)³
- Inner limit first. lim(x→2) (x²+1) = 4 + 1 = 5 (sum + power laws).
- Raise. 5³ = 125.
Your turn: limx→1 (x²+3)² = ?
Answer: 16.
Inner limit first: 1 + 3 = 4. Raise: 4² = 16.
Example 2 — linear inside: lim(x→3) (2x−1)²
- Inner limit: 2·3 − 1 = 5.
- Raise: 5² = 25.
Your turn: limx→4 (2x−3)³ = ?
Answer: 125.
Inner: 8 − 3 = 5. Raise: 5³ = 125.
Example 3 — combined with the quotient law: lim(x→1) [(x²+3)/(2x)]²
- Inner limit first (quotient law; denominator → 2 ≠ 0): (1+3)/(2·1) = 4/2 = 2.
- Raise: 2² = 4.
- Why this order: the power law applies to the whole fraction — evaluate the fraction’s limit, then square.
Your turn: limx→2 [(x²+1)/(3x)]² = ?
Answer: 25/36.
Inner limit (quotient law; denominator → 6 ≠ 0): (4+1)/6 = 5/6. Square: 25/36.
Example 4 — negative inside, even power: lim(x→−2) (x³+x)²
- Inner limit: (−2)³ + (−2) = −8 − 2 = −10.
- Raise: (−10)² = 100.
Your turn: limx→−3 (x²−x)² = ?
Answer: 144.
Inner: 9 − (−3) = 12. Raise: 12² = 144.
Memorization tips
- Inside first, power last: the four-word procedure. Say it while you work.
- The power law is the product law on autopilot: [f]n is f multiplied by itself n times — induction just writes that down formally.
- Integer n only: fractional exponents are the root law’s territory (domain checks required); negative exponents are the quotient law’s (needs L ≠ 0).
- It beats expanding: (2x−1)² → 25 via inside-first vs a FOIL expansion. Fewer steps, fewer sign errors.
- Watch the parity: even powers erase signs — (−10)² = 100. If your answer’s sign surprises you, check whether the exponent is even.
- Nesting order: with combined laws, always work inside-out: innermost limit, then the power, then the outer operation.
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 power law for limits?
If lim(x→a) f(x) = L exists and n is a positive integer, then lim(x→a) [f(x)]ⁿ = Lⁿ: evaluate the inside limit, then raise to the power.
Why is n restricted to positive integers?
Fractional powers can break on negative bases (√(−1) isn't real), and negative exponents are reciprocals — the quotient law's job, needing L ≠ 0. The integer-n version is always safe.
How does the power law relate to the product law?
It's the product law applied n times: [f]ⁿ is f multiplied by itself n times, so induction plus the product law gives Lⁿ. The constant multiple and sum laws often evaluate the inside.
Should I expand (x+1)² before taking the limit?
No need — evaluate the inside (3 at x→2) and square to get 9. The power law skips the expansion and its sign-error risk.
Does the power law work if the inner limit is negative?
Yes for integer n: lim(x→−2)(x³+x)² = (−10)² = 100. Only fractional powers of negative bases are problematic.
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