Answer by user361424 for Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$...
Either $L=-1$ or $L\neq-1$. First, let $L=-1$. Let $\epsilon=1$, $z=\frac\delta2$. Then:$$|z-0| = \frac\delta2 < \delta$$$$|f(z)-L| = |f(z)+1| = \left|\frac{\frac\delta2}{\frac\delta2}+1\right| = 2...
View ArticleAnswer by Yves Daoust for Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$...
With $z=e^{i\theta}$ we have$$\frac z{\bar z}=e^{2i\theta}=\cos2\theta+i\sin2\theta,$$ independently of $r$.Then as$$\left|\cos2\cdot0-\cos2\frac\pi2\right|=2,$$ for $\epsilon<1$, no $\delta$ can...
View ArticleAnswer by Angelo for Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$ doesn't...
We prove by contradiction that the limit $\;\lim_\limits{z \to 0} \frac{z}{\overline{z}}\;$ does not exist.If, by absurdum, the limit $\;\lim_\limits{z \to 0} \frac{z}{\overline{z}}$ existed, since...
View ArticleAnswer by heropup for Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$ doesn't...
The epsilon-delta argument can be made very simply, once you know that the limiting value is path-dependent. Let $$f(z) = z/\bar z = e^{2i\arg(z)}.$$ Then suppose there exists an $L \in \mathbb C$...
View ArticleProve $\lim_{z \to 0} \frac{z}{\overline{z}}$ doesn't exist using...
I'm trying to prove that the limit$$\lim_{z \to 0} \frac{z}{\overline{z}} \quad \qquad z \neq 0$$doesn't exist. Up to this point, the only definition of a limit for complex functions I know is as that...
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