Is a function differentiable in $x$ if $\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}=\infty$? All the definitions of differentiability I found (Wolfram Mathworld for instance) only require this limit to exist, but say nothing about the domain in which that has to happen.  
So what if that limit is $\pm\infty$? Wouldn't (in the "normal" interpretation of a function $\mathbb R \rightarrow \mathbb R$, for instance) the slope of the tangent of $f$ in $x$ be undefined?
Edit: a link clearing it up.

For a function $f: \mathbb{R} \to \mathbb{R}$ to be differentiable, we require that
$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = L,$$
where $L \in \mathbb{R}$.
Sometimes the wording is "the limit exists and is finite,"
but other times the finite portion is dropped, and the phrase "limit exists" means, in particular, that the limit is not $\infty$.
