In the lectures notes http://users.jyu.fi/~pkoskela/quasifinal.pdf (Prof. Koskela has made them freely available from his webpage, so I am guessing is OK that I paste the link here) Quasiconformality is defined by saying that $\displaystyle \limsup_\limits{r \rightarrow 0} \frac{L_{f}(x,r)}{l_{f}(x,r)}$ must be uniformly bounded in $x,$ where $\displaystyle L_{f}(x,r):=\sup_\limits{\vert x-y \vert \leq r} \{ \vert f(x)-f(y) \vert \}$ and $\displaystyle l_{f}(x,r):=\inf_\limits{\vert x-y \vert \geq r} \{ \vert f(x)-f(y) \vert \}.$
I have three questions concerning this definition:
1) The main question: When he proves that a conformal mapping is quasiconformal he says (at the beginning of page 5): "Thus, given a vetor h, we have that $|Df(x, y)h| = |∇u||h|$ By the complex differentiability of f we conclude that: $\limsup_\limits{r \rightarrow 0} \dfrac{L_{f}(x,r)}{l_{f}(x,r)}=1$"
And I don't quite understand how did he do that step. Is he perhaps using the mean value theorem and the maximum modulus principle?
2) Second question: Even accepting the previous argument, he only shows that conformal mappings are quasiconformal in dimension $2.$ How to do this in general? Also, is this definition the same if we replace $\vert x-y \vert \leq r$ and $\vert x-y \vert \geq r$ by $\vert x-y \vert =r$? The former bounds the latter trivially, but more than that I do not know.
3) What would be a nice visual interpretation of a quasiconformal mapping? How would look a map with possible infinite distortion at some points?
Thanks