Question

Calculate $\underset{x\to 1}{lim}(\frac{p}{1-x^p}-\frac{q}{1-x^q});p,q\in N$

Answer 1

Use formula $\underset{x\to a}{lim}\left(\frac{x^n-a^n}{x-a}\right)=n{\left(a\right)}^{n-1}$ to solve the problem.

$\underset{x\to 1}{lim}(\frac{p}{1-x^p}-\frac{q}{1-x^q})$

$=\underset{x\to 1}{lim}(-\frac{p}{(x-1)\frac{x^p-1}{x-1}}+\frac{q}{(x-1)\frac{x^p-1}{x-1}})$

$=\underset{x\to 1}{lim}(-\frac{p}{(x-1)p{\left(1\right)}^{p-1}}+\frac{q}{(x-1)q{\left(1\right)}^{q-1}})$

$=\underset{x\to 1}{lim}(-\frac{1}{(x-1)}+\frac{1}{(x-1)})$

$=\underset{x\to 1}{lim}\left(0\right)=0$