$\underset{x\to 0}{\lim }\frac{(\cos x{)}^{1/2}-(\cos x{)}^{1/3}}{{\sin }^{2}x}$ is

$\underset{x\to 0}{\lim }\frac{(\cos x{)}^{1/2}-(\cos x{)}^{1/3}}{{\sin }^{2}x}\left(\frac{0}{0}\text{ form }\right)$

$=\underset{x\to 0}{\lim }\frac{-\frac{1}{2}(\cos x{)}^{-1/2}\sin x+\frac{1}{3}(\cos x{)}^{-2/3}\sin x}{2\sin x\cos x}$

$=\underset{x\to 0}{\lim }\frac{-\frac{1}{2}(\cos x{)}^{-1/2}+\frac{1}{3}(\cos x{)}^{-2/3}}{2\cos x}$

$=-\frac{1}{2}\left[\frac{1}{2}-\frac{1}{3}\right]=-\frac{1}{12}$.