evaluation of recurring limit within square root.

Question

limit x tends to 1 $\left(1-\sqrt{x+1-\sqrt{\left(x-(1-\sqrt{\left(x+1-\sqrt{\left(x-(1-\sqrt{\left(x+.......∞\right)}\right)}\right)}\right)}}\right)^{\ln\left(\frac{1}{\left(x\right)}\right)}$
Is there any way to solve this complicated limit without L'Hôpital?
We can apply L'Hôpital by taking the logarithm on both sides, but how can the recurring limit be differentiated? Also if it had been a cube root or any other decimal power in place of square root, constant or varying throughout, then how would have been the approach? note--It is a 0^0 form, as ln1=0, and the recurring terms tend cut out to give 1, thus making the term tend towards 0. i tried using desmos for finite recurring and answer was 1. graph looked bit odd but continuous