1. Limit problems

Hello ,

I was solving problems on limits when i encountered this one with a complex number.

limit Z->0 (Z*/Z) WHERE Z*is conjugate of Z.

If i get 0/0 i used to factorize numerator, which almost always cancels with the denominator ,to get the answer which i could not do for this one.The problem for me is Z has two variables x and y and i dont know how to apply limits for them.I think the answer should be 'undefined' but i cant prove it mathematically.Any simple way to solve these kinds of problems?
Moreover are there any general ways of solving these limit problems?

Thank you

2. Originally Posted by essex
Hello ,

I was solving problems on limits when i encountered this one with a complex number.

limit Z->0 (Z*/Z) WHERE Z*is conjugate of Z.

If i get 0/0 i used to factorize numerator, which almost always cancels with the denominator ,to get the answer which i could not do for this one.The problem for me is Z has two variables x and y and i dont know how to apply limits for them.I think the answer should be 'undefined' but i cant prove it mathematically.Any simple way to solve these kinds of problems?
Moreover are there any general ways of solving these limit problems?

Thank you
Let z = x + iy.

Then you want $\lim_{(x, y) \rightarrow (0, 0)} \frac{x - iy}{x + iy}$.

Since the value depends on how the limit is taken, the limit does not exist.

3. Thank you mr fantastic .So we have to find the value as x approaches 0 (we get -1) and find the value as y approaches 0 (we get 1).Since both are different it doenot exist.

Is this the way we should do it for a number with different variables?
(i.e) by letting each variable of the number go to the given limit seperately and check if they are equal ?

4. Originally Posted by essex
Thank you mr fantastic .So we have to find the value as x approaches 0 (we get -1) and find the value as y approaches 0 (we get 1).Since both are different it doenot exist.

Is this the way we should do it for a number with different variables?
(i.e) by letting each variable of the number go to the given limit seperately and check if they are equal ?
You should consider different paths and see what happens. eg If $(x, y) \rightarrow (0, 0)$ along the path $y = mx$:

$\lim_{x \rightarrow 0} \frac{x - imx}{x + imx} = \lim_{x \rightarrow 0} \frac{1 - im}{1 + im} = \frac{1 - im}{1 + im}$ which clearly depends on $m$.

5. Oh ok. So it depends on the path given ,but if theres no specific path then we find it seperately for different variables and check if they are equal.correct?

Thank you very much