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?
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
WolframAlpha definitely says it's zero (and it even shows steps!):
Limit[(Re[z]Im[z])/ ;(|z|^2), z -> 0] - Wolfram|Alpha
Hmm...