1. ## Complex limit

I need to find this limit

$\displaystyle lim_{z\rightarrow{0}} \frac{Re(z)Im(z)}{|z|^2}$
Wolframalpha gives this limit as $\displaystyle 0$

Shouldn't this problem be equivalent

$\displaystyle lim_{(x,y)\rightarrow{(0,0)}} \frac{xy}{x^2+y^2}$
Wolframalpha gives this limit as non-existent
So now, I'm wondering, what makes these two formulations different?

2. ## Re: Complex limit

$\displaystyle \lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2}$ is ambiguous because the limit depends on what path you take to approach (0,0). For example, if you approach from the path y = x, you would obtain 1/2, but if you went from the path y = 0, the limit would be zero. Therefore the limit doesn't exist.

As for the first limit, I haven't studied complex limits but still, you know that z is a complex number but you have no idea which direction z is coming from in order to approach (0,0) or 0 + 0i. However, you know that

$\displaystyle Re(z) = |z|\cos{\theta}$ and $\displaystyle Im(z) = |z| \sin{\theta}$

so $\displaystyle \frac{Re(z)Im(z)}{|z|^2} = \frac{|z|^2 \sin{\theta} \cos {\theta}}{|z|^2} = \sin{\theta} \cos {\theta} = \frac{1}{2} \sin {2 \theta}$

Of course, theta is dependent on which path you are taking...if you are traveling on a straight line path towards the origin (e.g. $\displaystyle Im(z) = kRe(z)$), then theta would be constant and the limit would just be $\displaystyle \frac{1}{2} \sin{2 \theta}$

3. ## Re: Complex limit

Should be a bug at WolframAlpha, the limit does not exist.
I sent a bug report. It can take some months until they fix reported bugs, so I do not expect a response soon.

4. ## Re: Complex limit

WolframAlpha definitely says it's zero (and it even shows steps!):

Limit&#91;&#40;Re&#91;z&#93;Im&#91;z&#93;&#41;&#47 ;&#40;|z|&#94;2&#41;, z -> 0&#93; - Wolfram|Alpha

Hmm...

5. ## Re: Complex limit

Originally Posted by mfb
Should be a bug at WolframAlpha, the limit does not exist.
I think "z" is considered as real and in that case the limit is correct. And that is why the limit(x,y) is different.