Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 20 / c / ii / Solution

Near zero, $\cos x=1-x^2/2+O(x^4)$ and $\ln (1+v)=v+O(v^2)$, so $\ln \cos x=-x^2/2+O(x^4)$. Replacing $x$ by $3x$ similarly gives $\ln \cos 3x=-9x^2/2+O(x^4)$. ThereforeFor all sufficiently small nonzero $x$, both cosines are positive and the denominator logarithm is nonzero, so this quotient and its two-sided real limit are well-defined.