Existence of a complex sequence with given property

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

Mathematics Asked by Praveen on December 28, 2020

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

Answered by Kavi Rama Murthy on December 28, 2020

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