Suppose I define $\displaystyle g(x,y) = \frac {f(x)-f(y)}{x-y} $ and take $\displaystyle f(z) = z^{n} $. So $\displaystyle g(x,y) = x^{n-1}+ x^{n-2}y + \cdots + xy^{n-2} + x^{k-1} $. If I take $\displaystyle y=x $, then $\displaystyle g(x,x) = nx^{n-1} $. Is this valid? If I look at the original equation, does this also mean $\displaystyle g(x,x) = \frac {0}{0} $? I'm not sure how to understand division by zero.