Limits with complex numbers and their modulus

Question

If $z$ is a complex number, how would you determine the limit as $z$ tends to $0$ of $\ \frac{z^4}{|z|^4} $?

Answer 1

You are asking about the existence of $$ \lim_{z \to 0} \frac{z^4}{|z|^4}. $$

Let's first examine what happens if $z \to 0$ along the positive $x$-axis. Then, $z = x + 0i = x > 0$ and $x \to 0$, so the limit becomes $$ \lim_{x \to 0} \frac{x^4}{x^4} = 1. $$

Can we possibly approach along a different curve where you would get a different answer? For example, think about $\left(\sqrt{i}\right)^4 = -1$.