 Solution

For $x\ne 0$, the envelope ratio isIf $|x|\le 1$, use $|\sin x|\le |x|$ to bound this by $\frac{3}{2}(1+x^2)\le 3$. If $|x|\ge 1$, use $|\sin x|\le 1$ to bound it by $\frac{3}{2}(x^{-4}+x^{-2})\le 3$. The boundary equalities in these bounds cannot both occur in the sine estimate, so $M=3$ is a valid envelope. At zero the continuous ratio is $3/2$.

Generate independent Cauchy proposals as in (i), together with independent uniform $U$. Accept a proposal $Y$ whenusing the limiting value $1/2$ at $Y=0$. Repeat until acceptance. Part (a) proves exactness; the overall acceptance rate is $1/3$ and the mean number of proposals is three. No numerical maximization of the ratio is needed for correctness.