Existence of a complex sequence with given property

Question

How to show the existence of a complex sequence $(z_n)$ with $z_n\ne 1, \forall n$ but $\lim_{n\to\infty}z_n=1$ such that $\lim_{n\to \infty}\sin(\frac{1}{1-z_n})=100$? Can such a sequence be explicitly written?

Answer 1

Hint: Solve $w^{2}-200iw-1=0$. Then choose $\zeta_n \to \infty$ such that $e^{i\zeta_n} =w$ for all $n$. ( Take $c$ with $e^{ic}=w$ and take $\zeta_n =2n\pi +c$). Finally take $z_n=1-\frac 1{\zeta_n}$.