Edit: This question is wrong. Ignore it. I've already flagged to request for deletion. Please just go to that question: What exactly are the differences between real multivariable limits and complex limits?
Say you want to disprove the existence of either of the ff $\lim_{z \to 0} \frac{Re(z)}{|z|^2}$, $\lim_{z \to 0} \frac{Im(z)}{|z|^2}$ (as in here). It seems we just change the limits to $\lim_{(x,y) \to (0,0)} \frac{x \ \text{or} \ y}{x^2+y^2}$ and then go about this calc2 way.
- So the rule is that complex limit doesn't exist if the $\mathbb R^2$ limit doesn't exist? In general, for $$\lim_{z \to z_0}[u(z)+iv(z)] \ \text{vs} \ \lim_{(x,y) \to (x_0,y_0)}[u(x,y)+iv(x,y)],$$
where $u$ and $v$ are real functions, is it that the LHS doesn't exist if the RHS doesn't exist?
- Here, the RHS equals by theorem or by definition to $$\lim_{(x,y) \to (x_0,y_0)}u(x,y)+i\lim_{(x,y) \to (x_0,y_0)}v(x,y)$$
- BUT if the RHS exists, the LHS may or may not exist, i.e. existence of real limit is necessary but not sufficient for existence of complex limit? Please provide examples.
Note: For now, I'll just say the real functions $u,v$ without specifying specific domains and hope the above makes sense. If need be, then I can edit this question to be more specific about $u,v,z_0$, etc.
Related questions:
I've found several questions that talk about the relationship of complex derivative and real derivative, but what I'm not quite seeing is the general case/concept of complex vs real limits.
Differences between the complex derivative and the multivariable derivative.
Difference between the properties of differentiation in $\mathbb{C}$ and $\mathbb{R}^2$
Best Answer
Complex differentiation for $\mathbb C$ is defined differently than multivariate differentiation for $\mathbb R^2$. This is because you can divide by a complex number but not by a vector. It is NOT because limits work differently in the two spaces. In fact limits work identically in $\mathbb C$ and $\mathbb R^2$. More explicity, it is true that
$$ x_n + y_n i \to x + y i $$
as $n \to \infty$ if and only if
$$ \begin{pmatrix} x_n \\ y_n \end{pmatrix} \to \begin{pmatrix} x \\ y \end{pmatrix} $$
as $n \to \infty$.