Definition of conformal maps In the Conway's book "Functions of One Complex Variable", page 46, the definition of conformal map regards the angle preserving property plus the existence of the following limit:
$\lim\limits_{z\rightarrow a} \frac{|f(z)-f(a)|}{|z-a|}$.
What is the meaning of this additional limit property?
 A: Let $C_z$ be a curve in the complex z plane and $h = f(z)$ ananalytic function defining a change of variables from $z$ to $h$ the complex variable. Under this transformation the curve  $C_z$ is mapped into a new curve  $C_h$. We now look for geometrical properties of $C_h$ that are indemendent of the shape of $C_z$. The matematical description of $C_z$ can be given by $z(s) = x(s) + i y(s)$, with $s \in \mathbb{R}$. Suppose that $f(z)$ is analytic for $z$ in some domain of the complex $z$-plane denoted by D. The image $h$ of a continuous curve is also continuous. It can be written as $h = u(x(s),y(s)) + i v(x(s),y(s))$ with $u,v \in \mathbb{R}$. Because $x(s)$ and $y(s)$ are continuous functions of $s$, it follows that $u$ and $v$ are also continuous functions of $s$, which establishes the continuity of $h$. The part of $C_z$ that is contained in D is differentiable, and consequently its image is differentiable. In other word, the existance of
\begin{equation}
\lim\limits_{z\rightarrow a} \frac{|f(z)-f(a)|}{|z-a|} 
\end{equation}
guaranttee that $h$ is differentiable, but this is not enough. This is because $f(z)$ can map the part of $C_z$ contained in D into a curve which is self intersecting. In order to avoid this the additional conditions $f'(z_0) \neq 0$ and $z'(s_0) \neq 0$ are required. As a matter of fact, given
\begin{equation}
\frac{z(s)}{ds}=\frac{dx(s)}{ds}+i \frac{dy(s)}{ds}
\end{equation}
with $f(z)$ analytic in a domain containing the open neighborhood of $z_0 =z(s_0)$ we can write
\begin{equation}
\frac{h(s)}{ds}\big|_{s=s_0}=f'(z_0) \frac{dz(s)}{ds}\big|_{s=s_0}
\end{equation}
If $f'(z_0) \neq 0$ and $z'(s_0) \neq 0$ it follows that $h'(s_0)\neq 0$ and $\arg dh = \arg dz + \arg f'(z_0)$. For points where $f'(z_0) \neq 0$ , analytic transformations preserve angles.
