Limit WITHOUT using l'hospital

1. Jan 21, 2009

Doom of Doom

For $$p,q\in \mathbb{N}$$, determine $$\lim_{x\to1} \frac{x^{p}-1}{x^{q}-1}$$

I know that it should be p/q. Without using L'hospitals rule, how do I determine it?

2. Jan 21, 2009

chrisk

The quotient can be rewritten as (1-xp )/(1-xq ). Since x is less than 1 try a binomial expansion.

3. Jan 21, 2009

NoMoreExams

Factoring :)

4. Jan 21, 2009

Doom of Doom

Got it! My methods of trying were way more complicated. Don't know why I didn't think of that at first!

(x^n) - 1=(x-1)(x^(n-1) + ... + x + 1)

Thanks NoMoreExams