I had this simple looking limit:
$$\lim_{x\to 0} \frac{3^{(2^{x+1}-2)}-1}{\sqrt{1-\cos(x)}}$$
By some algebraic manipulation, I got here:
$$\frac{\sqrt{2}}{9}\lim_{x\to 0} \frac{3^{2^{x+1}}-9}{|\sin(x)|}$$
But now, I am quite unable to move further, the exponential tower is making a lot of mess and I don't see any way out of here. Using derivatives is prohibited. I would really appreciate if anyone could point me the right way.