Limit WITHOUT using l'hospital

[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?

 

[chrisk]

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

 

[NoMoreExams]

Factoring :)

 

[Doom of Doom]

Factoring :)

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