Does the limit $\lim_{x\to1^{-}}(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n}$ exist?

Let the lacunary power series $$f(x):=(1-x)\sum_{n=1}^{\infty}2^{n} x^{2^n},0<x<1.$$ We use Matlab to draw the function $f(x)$ graph, which shows that the function $f(x)$ has limits when $x\to1^{-}$. But we cannot rigorously prove it theoretically.