Evaluate the limit in $p$ and $q$ Evaluate $$\lim_{x \to 1}\left(\frac{p}{1-x^p}-\frac{q}{1-x^q}\right)$$ where $p,q$ are natural numbers. I have run out of ideas on how do I solve the limit. The few ideas I had did not work.

If you consider Laurent series built at $x=1$, you have $$\frac{1}{1-x^n}=-\frac{1}{n (x-1)}+\frac{n-1}{2 n}+\left(\frac{1}{12 n}-\frac{n}{12}\right)
   (x-1)+O\left((x-1)^2\right)$$ So $$\frac{p}{1-x^p}-\frac{q}{1-x^q}=\Big(-\frac{1}{x-1}+\frac{p-1}{2}+\left(\frac{1}{12}-\frac{p^2}{12}\right)
   (x-1)+O\left((x-1)^2\right) \Big)-\Big(-\frac{1}{x-1}+\frac{q-1}{2}+\left(\frac{1}{12}-\frac{q^2}{12}\right)
   (x-1)+O\left((x-1)^2\right)\Big)$$ that is to say $$\frac{p}{1-x^p}-\frac{q}{1-x^q}=\frac{p-q}{2}+\frac{q^2-p^2}{12} (x-1) +O\left((x-1)^2\right)$$ which gives not only the limit but also how it is approached.
