limn→∞ pn+qn2n, p,q>0 equals

$lim_{n \to \infty} {(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2})}^{n}, p, q > 0$ equals

A
1
B
$\sqrt{p q}$
C
$p q$
D
$\frac{p q}{2}$

Solution:

$lim_{n \to \infty} {(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2})}^{n}$

$= lim_{x \to 0} {(\frac{p^{x} + q^{x}}{2})}^{1 / x},$ where $n = \frac{1}{x}$

$= lim_{x \to 0} {[1 + \frac{1}{2} \{(p^{x} - 1) + (q^{x} - 1)\}]}^{1 / x} = \sqrt{p q}$

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