proving a limit of a function by definition

Question

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$

Prove that $$ \lim_{z \to i} f(z) = i $$ using the definition of the limit.

I have tried to find a relation between $\delta$ and $\epsilon$.

Answer 1

we have
$|z^3+2z| = |z-i||z^2+iz+1|=|z-i||(z-i+i)^2+i(z-i+i)+1|$
$=|z-i||(z-i)^2+2i(z-i)-1+i(z-i)-1+1| $
$ \le $$|z-i||z-i|^2+3|i||z-i|+1 $
$let $ $\delta = 1$
$|z^3+2z|$ $\lt $$\delta$$( 1^2+3 +1 ) =5\delta $

$Hence$ $\delta= min ${1,$\epsilon/5$ }$ $

I find it finally