Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 19

Since $A>0$ and $\omega >0$ are fixed, the standard sine limit givesIf $p>-1$, this tends to the assigned value zero. At $p=-1$, the right and left limits are $+A\omega$ and $-A\omega$; if $p<-1$, they diverge with opposite signs. HenceThe cancellation comes from the simple zero of $\sin (\omega x)$ at the origin, not rapid oscillation there.

Use the derivative definition and the specified value $f(0)=0$:The last factor tends to one. Therefore the limit is zero when $p>0$, equals $A\omega$ when $p=0$, and diverges to $+\infty$ when $p<0$. A finite derivative exists exactly whenIn particular $-1<p<0$ gives continuity without differentiability. The threshold is an example of regularity of a power times a function with a simple zero.

For every positive integer $n$, the bound $|\sin (\omega n)|\le 1$ givesFor $p<-1$, the positive p-series $\sum n^p$ converges. The comparison test therefore provesNo special arithmetic condition on $\omega$ is required for this sufficient condition.

Factor out the dominant cubic term and use a geometric series in $1/(3x^2)$:Thus the requested leading term and absolute error order areThe relative error of retaining only the leading term is $O(x^{-2})$; it differs from the absolute error order $O(x^{-5})$.