Solution
= Solution
For the final limit, near zero $\cos x>0$, so the real logarithm is well defined. First apply the proved rule to $\tan x/x$ on the right: its <derivative> ratio is $\sec^2x\to1$. Next apply it to $\log(\cos x)/x^2$; the <derivative> ratio is
$$
\frac{-\tan x}{2x}\longrightarrow-\frac12.
$$
The original quotient is even, so the left limit equals the right limit. Consequently
$$
\boxed{\lim_{x\to0}\frac{\log\cos x}{x^2}=-\frac12.}
$$