Past exam of the mathematical methods II course of the University of Oxford / 2014 / 2 / a / Solution

The Taylor expansions at zero give $1-e^x=-x-x^2/2+O(x^3)$ and $1-\cos x=x^2/2+O(x^4)$. Squaring the first expression gives $x^2+O(x^3)$, so canceling the leading quadratic power yieldsBoth numerator and denominator vanish, but their leading nonzero coefficients have ratio two.