Question Number 104856 by ~blr237~ last updated on 24/Jul/20

lim_(z→0)  (z^− /z)    ,    lim_(z→i)  (((z^− )^4 )/z^4 )  ,lim_(z→0)  ((sinz)/z)

Answered by abdomathmax last updated on 24/Jul/20

z =r e^(iθ)  z →0 ⇒r→0  lim_(z→o)  (z^− /z) =lim_(r→0)    ((re^(−iθ) )/(r e^(iθ) )) =lim_(r→0)   e^(−2iθ)   =e^(−2iθ)   but this limit is not unique ⇒the limit  dont exist  lim_(z→i)    (((z^− )^4 )/z^4 ) =(((−i)^4 )/i^4 ) =(−1)^4  =1  we have sinz =Σ_(n=0) ^∞  (((−1)^n  z^(2n+1) )/((2n+1)!)) ⇒  ((sinz)/z) =Σ_(n=0) ^∞  (((−1)^n  z^(2n) )/((2n+1)!)) =1−(z^2 /(2!)) +(z^4 /(5!)) −... ⇒  lim_(z→0)   ((sinz)/z) =1