Bit confused about a limit

I need to find this limit

Wolframalpha gives this limit as

Shouldn't this problem be equivalent

Wolframalpha gives this limit as non-existent

So now, I'm wondering, what makes these two formulations different?

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- June 13th 2012, 10:50 PMI-ThinkComplex limit
Bit confused about a limit

I need to find this limit

Wolframalpha gives this limit as

Shouldn't this problem be equivalent

Wolframalpha gives this limit as non-existent

So now, I'm wondering, what makes these two formulations different? - June 13th 2012, 11:42 PMrichard1234Re: Complex limit
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

and

so

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. ), then theta would be constant and the limit would just be - June 14th 2012, 09:56 AMmfbRe: 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. - June 14th 2012, 12:44 PMrichard1234Re: Complex limit
WolframAlpha definitely says it's zero (and it even shows steps!):

Limit[(Re[z]Im[z])/ ;(|z|^2), z -> 0] - Wolfram|Alpha

Hmm... - August 18th 2012, 11:15 PMTondaRe: Complex limit