I am working through the MIT 18.04 course notes (for fun, this is not a homework question) and I have a question about one of the "challenge" questions (for reference, it is on page 8 https://ocw.mit.edu/courses/mathematics/18-04-complex-variables-with-applications-spring-2018/lecture-notes/MIT18_04S18_topic2.pdf).
The challenge is to prove that $\lim_{h \to 0} \frac{\bar{h}}{h}$ can take any value with modulus 1, depending on the angle you approach $0$ with. I am having trouble parametrizing this. I can see a messy way to do it with a lot of triangles, but is there an easier way that I am missing?