Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 1 / 9D / iv / 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 ${\sec }^{2}x\to 1$. Next apply it to $\log (\cos x)/x^2$; the derivative ratio isThe original quotient is even, so the left limit equals the right limit. Consequently