Look at the change
Subtracting two nearby powers looks complicated. A local slope gives a simpler picture: change is approximately derivative times step.
A visual math lesson
A small step. A simpler difference.
Why a derivative reveals the leading term.
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What this lesson covers
- Read a finite difference as a slope multiplied by a step.
- Find the leading term of a difference of powers.
- Decide when a ratio has a finite, nonzero limit.
The core takeaway
For a fixed positive k and large n, nᵏ − (n − 1)ᵏ behaves like k · nᵏ⁻¹.
The lesson, in words
An equivalent explanation of the essential visual reasoning. Follow along, or read at your own pace.
The slope and the step
On the curve y = x to the power k, move from n − 1 to n. The horizontal step is 1. The derivative at n is k times n to the power k − 1. With fixed k > 0 and large n, multiplying this slope by 1 gives the leading change. Blue shows the curve; green identifies the tangent approximation; red marks the leading term.
nᵏ − (n − 1)ᵏ ≈ k · nᵏ⁻¹
Check the ratio
Here, approximately means that the ratio of the exact difference to the leading term tends to 1 as n increases. It is an asymptotic claim, not an exact equality for every n.
[nᵏ − (n − 1)ᵏ] / [k · nᵏ⁻¹] → 1
Why the smaller terms disappear
For a positive integer k, the binomial expansion cancels the two n to the power k terms. The first term left is k times n to the power k − 1. Each remaining term has a lower power of n, so its ratio to the leading term tends to zero.
Let the exponent decide
Consider n to the power 99 divided by the difference. Its leading behavior is (1/k) times n to the power 100 − k. For fixed k > 0, it tends to infinity when k < 100, to zero when k > 100, and to 1/100 when k = 100. The only finite nonzero case is k = 100.
n⁹⁹ / [nᵏ − (n − 1)ᵏ] ≈ (1/k) · n¹⁰⁰⁻ᵏ
Zoom in near zero
Rewrite the expression using t = 1/n. Near t = 0, the curve (1 − t) to the power k has tangent 1 − kt. Therefore 1 − (1 − t) to the power k is approximately kt. This also proves the leading-term result for any fixed real k > 0.
Use differentiation carefully
Repeated use of l’Hôpital’s rule requires checking its hypotheses at each step. Evaluating only k = 100 does not prove that no other k can give a finite nonzero limit. Comparing exponents handles all fixed positive k.
Keep the leading term
Difference is approximately derivative times step. The leading term is k times n to the power k − 1; the ratio in the example has a finite nonzero limit only at k = 100, where it equals 1/100.