Evaluate:$$\displaystyle \lim_{x \rightarrow 0}\frac{\cos(1-\frac{\sin x}{x})+\cos(2-\frac{\sin(2x)}{x})+\cdots+\cos(k-\frac{\sin(kx)}{x})-k}{\cos(1-\cos(x))+\cos(2-2\cos(2x))+\cdots+\cos(k-k\cos(kx))-k}$$
I can find this limit using L' Hospital Rule, I do not know how to do it without that.This question is proposed to Romanian Math Magazine by Jalil Hajimir