Question: Find the limit if any of 

$$ { \lim\_{z \rightarrow 3-4i}} \frac{(\bar z-3-4i)^4}{| z-3+4i|^4}$$

I have no idea how to start this question. Can anyone give a hint? Should i use L'hopital's Rule?

Answer: 
Yes it is ! Just take $$\varphi: \mathbb Z\longrightarrow M$$
defined by $$\varphi(m)= (m,-m).$$

The surjectivity is obvious. For the injectivity, it's easy to prove that $\ker \varphi=\{0\}.$
The pair $(m, -m)$ is an **element** of $\mathbb Z^2$

$\mathbb Z$ is a **set**, and with the operation $+$ it is a **group**.

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Speaking about homomorphisms between the two makes very little sensem because $(m, -m)$ is not a group (because even though technically, it is a set of two elements, you haven't defined an operation on it).

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What I feel you are **really** trying to ask is the following question:

> 
> Is the set $\{(m, -m)| m\in\mathbb Z\}$ isomorphic to $\mathbb Z$?
> 
> 
> 

In which case the answer is still *unclear*, because you **still** haven't defined an operation. Now, if the operation is $$(m, -m) + (n, -n) = (m+n, -(m+n)),$$

then **yes**, the two groups are isomorphic, and one possible isomorphism is ***the most obvious mapping you can possibly think of***.