<s>[INST] Solving a problem on limits 
$$\lim_{x\to 1}\left(\frac{p}{1-x^p}-\frac{q}{1-x^q}\right)  , p,q\in N$$

I tried fitting the formula $\lim_{x\to a}\frac{x^n - a^n}{x-a} = na^{n-1}$ but could not proceed further. Also I don't think that we should directly apply L'Hôpital's rule as it will be too lengthy. 
 [/INST] Using Taylor expansions around $0$, namely:


*

*$(1+u)^\alpha = 1+u+\frac{\alpha(\alpha-1)}{2}u^2 + o(u^2)$

*$\frac{1}{1+u} = 1-u+o(u)$ (a specific case of the first, for $\alpha=-1$)
You have, setting $h\stackrel{\rm def}{=} 1-x \xrightarrow[x\to1]{}0$ (to center things around $0$, with which I am much more familiar than around $1$):
$$\begin{align}
\frac{p}{1-x^p}-\frac{q}{1-x^q}
&= \frac{p}{1-(1-h)^p}-\frac{q}{1-(1-h)^q}
\\
&= \frac{p}{1-(1-ph+\frac{p(p-1)}{2}h^2+o(h^2))}-\frac{q}{1-(1-qh+\frac{q(q-1)}{2}h^2+o(h^2))}\\
&= \frac{p}{ph-\frac{p(p-1)}{2}h^2+o(h^2)}-\frac{q}{qh-\frac{q(q-1)}{2}h^2+o(h^2)}\\
&= \frac{1}{h-\frac{p-1}{2}h^2+o(h^2)}-\frac{1}{h-\frac{q-1}{2}h^2+o(h^2)}\\
&= \frac{1}{h}\left(\frac{1}{1-\frac{p-1}{2}h+o(h)}-\frac{1}{1-\frac{q-1}{2}h+o(h)}\right)\\
&= \frac{1}{h}\left(1+\frac{p-1}{2}h+o(h)-(1+\frac{q-1}{2}h+o(h))\right)\\
&= \frac{1}{h}\left(\frac{p-1-(q-1)}{2}h+o(h)\right)\\
&= \frac{p-q}{2}+o(1) 
\xrightarrow[h\to0]{}\boxed{\frac{p-q}{2}}
\end{align}$$
</s>