Let $x\in (0,1)$, $ M(v,x) = \bigl(1+v^x\bigr) \bigl(v^{1/x}-v\bigr) + x \bigl(1+v^{1/x}\bigr) \bigl(v^x-v\bigr) $ and let $v_0(x)$ be the root of $M(v,x)$ in $(0,1)$. As $x \rightarrow 1$ this root (numerically) approaches a value $0.090776278\ldots$ which seems to be $(\alpha-1)/(\alpha+1)$ where $\alpha = 1.199678\ldots$ is the positive solution to $\coth(\alpha) = \alpha$, a quantity related the the Laplace limit constant (see A033259 and the references therein). I'm having problems proving this, any pointers would be appreciated.
edit: changed $[0,1]$ to $(0,1)$ as the location of the root of $M$: as mentioned in the original version of joriki's answer, both $0$ and $1$ are roots of $M$ (independent of $x$), so the former implies that $v_0$ is not well-defined.