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  4. The value of displaystyle lim x arrow 1((p/1-xp)-(q/1-xq)), p, q, ∈ N, equals

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Q. The value of $\displaystyle\lim _{x \rightarrow 1}\left(\frac{p}{1-x^{p}}-\frac{q}{1-x^{q}}\right), p, q, \in N$, equals

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 1} \frac{p-q+q x^{p}-p x^{q}}{1-x^{p}-x^{q}+x^{p+q}} \left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 1} \frac{p q x^{p-1}-p q x^{q-1}}{-p x^{p-1}-q x^{q-1}+(p+q) x^{p+q-1}}\left(\frac{0}{0}\right)$ ($L’$ Hospital’s rule)
$=\displaystyle\lim _{x \rightarrow 1} \frac{p q(p-1) x^{p-2}-p q(q-1) x^{q-2}}{-p(p-1) x^{p-2}-q(q-1) x^{q-2}+(p+q)(p+q-1) x^{p+q-2}}$ ($L’ $ Hospital’s rule)
$=\frac{p-q}{2}$