Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 15 / c / ii / Solution

Both numerator and denominator vanish quadratically. Factor out $x^2$ before dividing the power series:The reciprocal of the denominator is $1+x^2/2-x^4/12+O(x^6)$. HenceThe expansion describes the continuous extension at the removable singularity $x=0$, where the original quotient is undefined but its limit is one. Odd powers vanish because the quotient is even.