Evaluate $\lim\limits_{x\to\ 0}\frac{1-\cos x\times (\cos2x)^{\frac{1}{2}}\times (\cos3x)^{\frac{1}{3}}}{x^2}$ without L'Hospital's rule I have no idea how to solve this limit. I tried transforming the expression
$$\frac{1-\cos x\times (\cos2x)^{\frac{1}{2}}\times (\cos3x)^{\frac{1}{3}}}{x^2}$$
in
$$e^{\ln\frac{1-\cos x\times (\cos2x)^{\frac{1}{2}}\times (\cos3x)^{\frac{1}{3}}}{x^2}}$$
but that didn't work. 

You can write $$1-\cos x(\cos 2x)^{1/2}(\cos3x)^{1/3}=1-\cos x+\cos x(1-(\cos 2x)^{1/2}(\cos3x)^{1/3})=$$$$1-\cos x+\cos x\left((1-(\cos 2x)^{1/2}+(\cos 2x)^{1/2}\left(1-(\cos3x)^{1/3}\right)\right),$$ therefore $$\frac{1-\cos x(\cos 2x)^{1/2}(\cos3x)^{1/3}}{x^2}=$$$$\frac{1-\cos x}{x^2}+\cos x\frac{1-(\cos 2x)^{1/2}}{x^2}+\cos x(\cos 2x)^{1/2}\frac{1-(\cos3x)^{1/3}}{x^2}.$$ But, for all $n\in\mathbb N$, $$\frac{1-(\cos nx)^{1/n}}{x^2}=\frac{1}{1+(\cos nx)^{1/n}+(\cos nx)^{2/n}+\dots+(\cos nx)^{1-1/n}}\frac{1-\cos nx}{x^2}=$$$$\frac{n^2}{1+(\cos nx)^{1/n}+(\cos nx)^{2/n}+\dots+(\cos nx)^{1-1/n}}\frac{1-\cos nx}{(nx)^2}\xrightarrow{x\to 0}\frac{n^2}{2n}=\frac{n}{2}.$$ Therefore, the requested limit is equal to $$\frac{1}{2}+\frac{2}{2}+\frac{3}{2}=3.$$
