How to find $\arg\lim_{n \to \infty}(1+\frac{z}{n})^n$?

Question

$z\in \mathbb{C}$.

The equality $\lim_{n \to \infty}(1+\frac{z}{n})^n=e^z$ should not be used.

Answer 1

$$Arg\left[\left(1+\frac{z}{n}\right)^n\right]=nArg\left(1+\frac{z}{n}\right)$$

$$1+\frac{z}{n}=1+\frac{x}n+i\frac{y}n\implies Arg\left(1+\frac{z}{n}\right)=\arctan\frac{y}{n+x}$$

then

$$Arg\left[\left(1+\frac{z}{n}\right)^n\right]=nArg\left(1+\frac{z}{n}\right)=n\arctan\frac{y}{n+x}\sim \frac{ny}{n+x}\to y$$