Đặt f(x)=\frac{x^{\alpha }-a^{\alpha }}{x^{\beta }-a^{\beta }}Ta có f(x)=\frac{(x-a)(x^{\alpha -1}+x^{\alpha -2}a+....+x.a^{\alpha -2}+a^{\alpha -1})}{(x-a)(x^{\beta -1}+x^{\beta -2}a+....+x.a^{\beta -2}+a^{\beta -1})}
= \frac{x^{\alpha -1}+x^{\alpha -2}a+....+x.a^{\alpha -2}+a^{\alpha -1}}{(x^{\beta -1}+x^{\beta -2}a+....+x.a^{\beta -2}+a^{\beta -1}}
Cho x_{n}=a ta có f(x)=\frac{\alpha .a^{\alpha -1}}{\beta .a^{\beta -1}}=\frac{\alpha }{\beta }. a_{\alpha -\beta }
Ta có
A= limxα−aαxβ−aβx→a = lim (\frac{\alpha }{\beta}.a_{\alpha -\beta }α−aαxβ−aβ