1) Prove that $\displaystyle \lim_{z\to0}z+|z|^3$ exists and equals $\displaystyle 0$.

2) Prove by letting $\displaystyle z$ approach $\displaystyle 0$ along suitable rays that $\displaystyle \lim_{z\to0}{\bar{z}\over z}$ fails to exist.

The only thing I can use the $\displaystyle \epsilon-\delta$ definition.

I have a few like each of these to do, but no examples were given.