Question Number 108678 by Rasikh last updated on 18/Aug/20
Answered by abdomsup last updated on 18/Aug/20
$${dont}\:{exist}\:\:{let}\:{x}_{{n}} =\left(\frac{\mathrm{1}}{{n}}\right)^{{i}} \\ $$$${lim}_{{n}\rightarrow+\infty} {x}_{{n}} \:\:={o}^{{i}} \:\:{but} \\ $$$${x}_{{n}} ={e}^{{iln}\left(\frac{\mathrm{1}}{{n}}\right)\:={e}^{−{iln}\left({n}\right)} \:={cos}\left({lnn}\right)+{isin}\left({lnn}\right)} \\ $$$${but}\:{cos}\left({lnn}\right)\:{and}\:{sin}\left({lnn}\right)\:{havent} \\ $$$${any}\:{limit} \\ $$$$ \\ $$
Answered by Her_Majesty last updated on 18/Aug/20
$${z}={re}^{{i}\theta} \wedge{r}\geqslant\mathrm{0}\wedge\theta\in\mathbb{R}\:\Rightarrow\:{z}^{{i}} ={r}^{{i}} {e}^{−\theta} =\frac{\mathrm{1}}{{e}^{\theta} }{e}^{{ilnr}} \\ $$$${this}\:{is}\:{defined}\:{for}\:\theta\in\mathbb{R}\wedge{r}\neq\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\mathrm{0}^{{i}} \:{is}\:{not}\:{defined} \\ $$