My question is somewhat simple I think, I just want to know to treat the combinatorial part in the expression for $$\lim_{x \to y} \frac{x^{n}-y^{n}}{x-y}$$.
What I mean by this is I know how to solve this question but I was attempting to do it with another notation. So normally what transpires is the following:
$$\lim_{x \to y} \frac{x^{n}-y^{n}}{x-y} \\ = \lim_{x \to y} \frac{(x-y)(x^{n-1}+yx^{n-2}+ \dots y^{n-1})}{x-y} = ny^{n-1}$$
But I was attempting to express the larger polynomial as $$(x^{n-1}+yx^{n-2}+ \dots y^{n-1})= (x+y)^{n-1} = \sum_{m = 0}^{n-1}\binom{n-1}{m}x^{(n-1)-m} y^{m}$$.
Doing this I run into problems with the combinatorial expression for the coefficients. How would I remedy this? I know I was probably trying to be too clever for my own good, but compact notation is what is desired at a higher level so I may as well get practice in now.
Just posting my comment as an answer.
Note that $$ (x + y)^{n-1} = x^{n-1} + (n-1)x^{n-2} y + \ldots + y^{n-1} = \sum_{k = 0}^{n-1} {n-1 \choose k} x^{(n-1) - k} y ^k $$ But $$ x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + x^{n-3} y + \ldots + y^{n-1}) = (x-y) \left( \sum_{k = 0}^{n-1} x^{(n-1) - k} y^k \right) $$