Cute Limit

$\lim_{x\to1} \frac{\sin(1-\cos(x-1))}{x(x-1)} \cdot \frac{1-\cos(x-1)}{1-\cos(x-1)}$ $=\lim_{x\to1} \frac{1-\cos(x-1)}{x(x-1)}$ You can make a change of variable for seeing this a simpler way: $u=x-1$ . $\lim_{u\to0} \frac{1-\cos(u)}{u(u+1)} = \lim_{u\to0} \frac{1-\cos(u)}{u(u+1)} \cdot \frac{1+\cos(u)}{1+\cos(u)}$ $\lim_{u\to0} \frac{1-\cos^2(u)}{u(u+1)(1+\cos(u))} = \lim_{u\to0} \frac{\sin(u)}{u} \cdot \frac{\sin(u)}{(u+1)(1+\cos(u))}=0$