$\exp\lim_{x \to 1}\ln ((1+x)/(2+x))^{(1-\sqrt{x}/1-x)} $
$\exp \lim_{x\to 1 }((1-\sqrt{x})/(1-x))\ln ((1+x)/(2+x))$
$(1-\sqrt{x}/1-x)$ forms 0/0 forms when x=1 we can apply LHospital rule here So,
$\exp((-0.5)*(x\tfrac{-3}{2})/-1)*\ln (2/3)$
Putting the value of x=1
$\exp \ln (2/3)^{0.5}$
Answer will be$\sqrt{2/3}$