Answer
$\lim\limits_{x \to \infty}x^{(ln~2)/(1+ln~x)}=2$
Work Step by Step
$\lim\limits_{x \to \infty}x^{(ln~2)/(1+ln~x)}=\lim\limits_{x \to \infty}(e^{ln~x})^{(ln~2)/(1+ln~x)}=\lim\limits_{x \to \infty}e^{(ln~x)(ln~2)/(1+ln~x)}$
$\lim\limits_{x \to \infty}\frac{(ln~x)(ln~2)}{1+ln~x} = \lim\limits_{x \to \infty}\frac{(ln~2/x)}{(1/x)} = ln~2$
Therefore:
$\lim\limits_{x \to \infty}e^{(ln~x)(ln~2)/(1+ln~x)} = e^{ln~2} = 2$