Question

The value of $\underset{x\to 1}{lim}\left(\frac{p}{1-x^p}-\frac{q}{1-x^q}\right),p,q,\in N$, equals

• A. $\frac{p+q}{2}$
• B. $\frac{pq}{2}$
• C. $\frac{p-q}{2}$
• D. $\sqrt{\frac{p}{q}}$

Answer 1

The correct option is C $\frac{p-q}{2}$

$\underset{x\to 1}{lim}\left(\frac{p}{1-x^p}-\frac{q}{1-x^q}\right)$
$\underset{x\to 1}{lim}\frac{p-q+q{x}^p-p{x}^q}{1-x^p-x^q+x^{p+q}}....\left(\frac{0}{0}form\right)$
$=\underset{x\to 1}{lim}\frac{pq{x}^{p-1}-pq{x}^{q-1}}{-p{x}^{p-1}-q{x}^{q-1}+(p+q)x^{p+q-1}}\left(\frac{0}{0}\right)\text{(using L'Hospital's rule)}$
$=\underset{x\to 1}{lim}\frac{pq(p-1)x^{p-2}-pq(q-1)x^{q-2}}{-p(p-1)x^{p-2}-q(q-1)x^{q-2}+(p+q)(p+q-1)x^{p+q-2}}$$\text{(L'Hospital's rule)}$
$=\frac{p-q}{2}$