Evaluate $\lim_{x\to 1} \frac{p}{1-x^p}-\frac{q}{1-x^q}$

Symmetry! Let $f_{p,q}(x) = \frac{p}{1-x^p}-\frac{q}{1-x^q}$. Then: $$ f_{p,q}\left(x^{-1}\right) = (p-q)-f_{p,q}(x)$$ hence, assuming that the original limit exists:

$$ \lim_{x\to 1}f_{p,q}(x) = \frac{1}{2}\lim_{x\to 1}\left(f_{p,q}(x)+f_{p,q}(x^{-1})\right) = \frac{1}{2}\lim_{x\to 1}(p-q)=\color{red}{\frac{p-q}{2}}.$$

The assumption $p,q\in\mathbb{N}^+$ is in fact unnecessary, we just need $p,q\neq 0$.