Write $z = re^{i\theta}$, so $\bar{z} = re^{-i\theta}$ and $\bar{z}/z = e^{-2i\theta}$. As $z\to 0$ along different paths (different $\theta$), the limit takes different values. For example, along the real axis ($\theta=0$): limit $= 1$; along the imaginary axis ($\theta=\pi/2$): limit $= e^{-i\pi} = -1$. Since the limit depends on the path, $\displaystyle\lim_{z\to 0}\frac{\bar{z}}{z}$ does not exist.