I just watched a video lecture where it's proved that $\lim_{z\rightarrow 0}{\frac{\bar{z}}{z}}$ does not exist: by putting the condition that $z=x+iy$ and $y=\lambda x$, it comes out that $\frac{\bar{z}}{z}=\frac{1-i\lambda}{1+i\lambda}$, which depends on $\lambda$, hence the limit doesn't exist.
However WolframAlpha says it's equal to 1. Why? Is it making some more advanced assumptions? Is the proof above correct?