Root Limit

Let $x=nt.$ Divide through by $n^x$ to get $\left( 1-\frac1{n} \right)^{nt} + \left( 1-\frac2{n} \right)^{nt} = 1.$ As $n\to\infty,$ this becomes $\frac1{e^t} + \frac1{e^{2t}} = 1,$ which simplifies to $e^{2t} - e^t - 1 = 0,$ so $e^t = \frac{1+\sqrt{5}}2$ (the negative sign doesn't work because $e^t$ is positive), so $t = \ln\left( \frac{1+\sqrt{5}}2 \right) \approx \fbox{0.4812}.$

This is a bit hand-wavy, of course!