I am kinda of new to the methods of proving limits with the epsilon delta method, and I can figure out simple limits, but more general ones seem to be quite difficult. Heres the question at hand:

Prove, by epsilon-delta methods, the limit of the following:

$\displaystyle \lim_{x \to y} \frac{x^n -y^n}{x-y}$

I know the rules for minipulating limits, as in "the limit of a sum of functions is the sum of the limits of those functions" and "the limit of the product of two functions is the product of the limits of those functions", ect. I've gone at this problem and haven't had much luck, I've realized that you can factor out a $\displaystyle (x-y)$ from the top, leaving the second factor of the expanded factorization of $\displaystyle x^n - y^n$ next to it. But, I'm not sure if I should cancel the $\displaystyle (x-y)$'s because I feel like they will be crucial and usefull in implying $\displaystyle |\frac{x^n - y^n}{x-y} - L|< \epsilon$ from $\displaystyle 0 < |x-y| < \delta$. Any help would be much appriciated, and also if anybody can give a general method of how to tackle general limits like this one, I would appreciate that also.