Proof
- For a whole number n: xⁿ − aⁿ = (x − a)(xⁿ⁻¹ + xⁿ⁻²a + … + aⁿ⁻¹).
- Cancel (x − a). The bracket has n terms, and each tends to aⁿ⁻¹ as x → a.
- So the limit is n aⁿ⁻¹. It holds for any rational n too.
Standard limits
Questions students ask
What is the limit of (xⁿ − aⁿ) / (x − a) as x tends to a?
n aⁿ⁻¹.
How do you prove lim x→a (xⁿ − aⁿ) / (x − a) = n aⁿ⁻¹?
For a whole number n: xⁿ − aⁿ = (x − a)(xⁿ⁻¹ + xⁿ⁻²a + … + aⁿ⁻¹). Cancel (x − a). The bracket has n terms, and each tends to aⁿ⁻¹ as x → a. So the limit is n aⁿ⁻¹. It holds for any rational n too.