Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 24F / d / Solution

If $f_0(z)=i\mathrm{Log}z$ is the projection from part (b), then any other Mercator projection $f$ gives an automorphism $f\circ f_0^{-1}$ of $E$ preserving its two ends. Part (c) therefore yieldsEvery map in this family plainly has the required limiting behaviour.