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I am trying to find if $$\lim_{r\to0}\frac{r(\cos^2\theta-\sin^2\theta) + i|r|\sqrt{|\cos\theta\sin\theta|}}{e^{i\theta}}$$ has a limit.

Does the limit tend to $0$ as $r\to0$? Because the numerator will be $0$?

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up vote 1 down vote accepted

This limit does exist. But if this is supposed to be following up on this question, then what you need is $$\lim_{r\to0}\frac{r(\cos^2\theta-\sin^2\theta) + i(|r|/r)\sqrt{|\cos\theta\sin\theta|}}{e^{i\theta}},$$ with $|r|/r$ rather than just $r$ in the second term in the numerator. Then notice that $|r|/r=1$ if $r>0$ and $=-1$ if $r<0$. The first term then approaches $0$ as $r\to0$, but the second term approaches different limits as $r\to0+$ and as $r\to0-$. So the limit you would be looking for does not exist because the two one-sided limits differ (except when $\cos\theta\sin\theta=0$).

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