Tell me, Do you? (Version 2)

We can see this limit as a derivative:

$\lim_{h\to 0} \frac{(1+h)^{\cos(h^2)}-1}{h} =\frac{d}{dx}\left(x^{\cos(x-1)^2}\right)|_{x=1} =\frac{d}{dx}\left(e^{\cos(x-1)^2\ln x}\right)|_{x=1}$ $=e^{\cos(x-1)^2\ln x}\cdot (\cos(x-1)^2\frac1x-\sin(x-1)^2\cdot\ln x \cdot 2(x-1)) |_{x=1}$ $=1(1-0)=\boxed{1}.$