Calculating $\lim_{x \rightarrow 1}(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}})$ How to calculate following limit?
$$\lim_{x \rightarrow 1}\left(\frac{23}{1-x^{23}} - \frac{31}{1-x^{31}}\right)$$

Alternatively, applying L'Hopital's rule, we get:
\begin{align}
\lim_{x \to 1}{\frac{23}{1-x^{23}}-\frac{31}{1-x^{31}}}& =\lim_{x \to 1}{\frac{23(x^{23}+x^{24}+ \ldots +x^{30})-8(1+x+\ldots +x^{22})}{(1-x^{31})(1+x+ \ldots +x^{22})}} \\
& =\lim_{x \to 1}{\frac{23(23x^{22}+24x^{23}+ \ldots +30x^{29})-8(1+ 2x+\ldots +22x^{21})}{-31x^{30}(1+x+ \ldots +x^{22})+(1-x^{31})(1+2x+\ldots+22x^{21})}} \\
& =\frac{23(22(8)+\frac{8(9)}{2})-8(\frac{22(23)}{2})}{-31(23)} \\
& =-4
\end{align}
